Physical Metallurgy Principles SI Version

Transcription

1 Fourth Edition Physical Metallurgy Principles SI Version Chapter Eight: Annealing 8-1

2 8.1 Stored Energy of Cold Work Strain hardening (work hardening) is the phenomenon whereby a ductile metal becomes harder and stronger as it is plastically deformed. The temp. at which deformation takes place is cold relative to the melting temp. of the metal. Cold working 2

3 Energy put into cold work Most of the energy expended in cold work appears in heat. A finite fraction is stored as strain energy associated with various lattice defects created by the deformation (ex. dislocations, point defects) rate (%): decreasing with deformation (strain) (5%) (amount due to deformation) (6 calories, or 25 J/mole) 3

4 The nature of stored energy of plastic deformation 1. cold working is able to increase the number of dislocations significantly (~ m -2 ) (annealed sample ~ m -2 ) 2. point defects: another source of retained energy => one mechanism has been described (dislocation intersection: jogs: either vacancies and interstitials as it glides) 4

5 Jogs (Edge with a jog) extra plane (screw with a jog) Chap. 4 mn, no, and op move over a stepped surface (not gliding along a plane) (extra plane) b 5

6 Supplement (the principles of engineering materials, p. 294) The overall elastic strain energy or free energy of the material increases with increasing plastic strain (increasing dislocation density). As the temp. is increased, thermal energy aids dislocation movement (both glide and climb) so that dislocations may begin to move in response to the stress fields of 6 surrounding dislocations.

7 8.2 The Relationship of Free Energy to Strain Energy The free energy of deformed metal is greater than that of an annealed metal by an amount approximately equal to the stored strain energy. => minimize dislocations for annealed metal. => G = H (H = U + constant P and V) G: free energy associated with the cold work H: the enthalpy (or the stored strain energy) S: the entropy increase due to the cold work The free energy increase equated directly to the stored energy 7

8 High free energy for the cold worked materials => soften (less dislocations, deformation) spontaneously. A metal does not usually return to the annealed condition by a single simple reaction because of the complexity of the cold-worked state. Heating a deformed metal, greatly speeds up its return to the softened state. 8

9 8.3 The Release of Stored Energy Valuable info about the nature of the reactions that occur as a cold-worked metal returns to its original state may be obtained through a study of the release of its stored energy. There are two important methods of accomplishing this: 1. Anisothermal anneal: sample is heated from low to high temperature; the energy release is determined as a function of temperature. 9

10 Energy release recrystallization recovery I II grain growth III 2. Isothermal anneal: the free energy is measured while the specimen is maintained at a constant temp. Energy release recrystallization recovery I II III grain growth 10

11 These large energy releases appear simultaneously with the growth of an entirely new set of essentially strain-free grains, which grow at the expense of the original badly deformed grains. May be understood as a realignment of the atoms into crystals with a lower free energy The three stages of releasing stored energy: recovery, recrystallization, and grain growth. 11

12 Cold-worked 300 C 370 C 410 C 460 C 650 C 12

13 8.4 Recovery: Working increases strength, hardness, and electrical resistance, and it decreases ductility. Laue patterns of deformed single crystals show pronounced asterism corresponding to lattice curvatures. Debye-Scherrer (powder): broaden diffraction lines In the recovery stage of annealing, the cold-worked crystal will tend to restore its original physical and mechanical properties. The various physical and mechanical properties do not recover their values at the same rate, indicating the complicated nature of the recovery process. Some properties are more effectively restored at different stages of annealing; e.g. hardness is insensitive to recovery but sensitive to recrystallization. 1

14 Another anisothermal anneal curve Almost no recovery Recover a lot recrystallization recovery I II grain growth III The fraction of the energy released during recovery is much larger than Fig

15 8.5 Recovery in Single Crystals (simple form of plastic deformation) The complexity of the cold-worked state is directly related to the complexity of the deformation that produces it. For a single crystal: lattice distortions are simpler by easy glide (slip on a single plane) than by multiple glide (simultaneous slip on several systems). Recovery associated with a simple form of plastic deformation: the simple recovery process involves the annihilation of excess dislocations. (positive and negative edge, left and right screw annihilation). it is probable that both slip and climb mechanism are involved. For a polycrystalline metal: lattice distortion may be more severe. 3

16 if the deformation does not involve bending for a single crystal (easy glide) => it is quite possible to completely recover its hardness without recrystallization. => however, in generally, deformation by easy glide could not be removed even at temperatures close to melting point. compressive Neutral axis Max. tensile Zero Shear stress Slip planes Max. 4

17 Without loss of time 1 st load 2 nd load 3 rd load Completely recovery Recovery begins very rapidly. The rate at which a property recovers isothermally is a decreasing function of the time. 5

18 quicker Zn single crystal: deformed by easy glide at 223K Shows the time required to recovery for different temp. The rate of recovery is much faster at 283K. The temp. dependence of recovery time is a Arrhenius-type behavior (easy glide). : the time required to recover a given fraction of the total yield point R 6

19 2 e e Q / RT Q / RT Q 1 1 ( ) R T T e Q: J/mole for Zn If for recovery of ¼ of its original yield point in 5 min at 273K (0 C) T 2 = 273 K 2 = 5 min ( e ( e ) ) 0.185min At 300K (27 C) min(12.5days) At 223K (-50 C) 7

20 8.6 Polygonization: much more complicated process Another recovery process is called polygonization => in its simplest form, it is associated with crystals that have been plastically bent. => one of the most important recovery processes. Polygonization 1

21 Driving force for recovery: the reduction in cold worked strain energy. Relieving the strain energy. Rearranging these dislocations to produce a subgrain structure. The development of this configuration which is decidedly of lower energy is known as polygonization. 2

22 In a Laue photograph, for a single crystal: A finite number of spots is obtained. Each of the elongated, or asterated spots of the deformed crystal is observed before anneal. Crystals with nearly identical orientations (low-angle boundary, low-energy dislocation structure), their patterns will almost coincide. 3

23 Strain field: additive Active planes Strain field: partially cancel each other A plastically bent crystal must contain an excess of positive edge dislocations that lie along active planes (additive). => high strain energy 4

24 Strain field: partially cancel each other Strain field: additive In addition to lowering the strain energy, the regrouping of edge dislocations into low-angle boundaries has a second important effect. => removal of general lattice curvature. 5

25 It is customary to call low-angle boundaries, such as develop in polygonization, subboundaries. The dislocation introduced during working are arranged into more stable configurations (lower strain energy). (the principles of engineering materials, p. 296) 6

26 8.7 Dislocation Movements in Polygonization An edge dislocation: vertical movement: climb; horizontal movement: slip Both are required in polygonization 7

27 Dots: dislocations Associated with slip planes Various stages of polygonization subboundaries Y- junction Further coalescence Coalescence of the low-angle boundaries All the dislocations lie in the subboundaries (low angle boundaries, or polygon walls) Polygonization process by a simple bending and annealed for 1 hr for ironsilicon single crystals. Combine the pairs of branches into single polygon walls. 8

28 BCC previous photographs. Photograph of the front surface 9

29 More complicated case: polycrystalline slip occurs on a number of intersecting slip planes, and lattice curvatures are more complex and vary with position in the crystal. Such complex deformation even happens for single crystals. 10

30 400 C 600 C 800 C Typical substructure by TEM (A) high density of dislocations, with no well defined cell boundaries. (B) dislocation density, and random arrays of dislocations are visible (1280 min). 11

31 The formation of these dislocation arrangements produces only a slight decrease in hardness, since the dislocation density changes very little. Electrical properties are recovered. (the principles of engineering materials, p. 296) 12

32 8.8 Recovery Processes at High and Low Temperatures Polygonization is too complicated a process to be expressed in terms of a simple rate equation (slide 18). (not as easy as slip). Polygonization involves both slip and climb. Relatively high temp. (because of climb) is required for rapid polygonization. 13

33 8.9 Recrystallization Energy release recrystallization recovery I II III grain growth Energy detected by a microcalorimeter (13mJ/hr) 14

34 Recovery competes with recrystallization, as both are driven by the stored energy. also thought to be a necessary prerequisite for the nucleation of recrystallized grains. 15

35 8.10 The Effect of Time and Temp. on Recryslallization One way to study the recrystallization process is to plot isothermal recrystallization curves like The temp., the time needed to finish the recrystallization. 16

36 (1/T) Empirical equation: 1/T = K Log + C K: slope, C: intercept, 1/ : the rate at which 50% of the structure recrystallized 17

37 Q R for activation enthalpy for the motion of vacancies: a simple physical properties: barrier height. But, the physical significance of Q R for recrystallization is not completely understood. But, couple good reasons 18

38 3. Although recrystallization tends to follow a pattern like a nucleation and growth, the formation of a nucleus does not occur. The origin of a recrystallized grain is always a preexisting region that is highly misoriented in relation to the material surrounding it. This high degree of misorientation gives the region from which the new grain originates the needed growth mobility. 19

39 8.11 Recrystallization Temperature Recrystallization temperature: the temperature at which a particular metal with a particular amount of cold work will completely recrystallize in a finite period of time (1hr). Because of the large Q R (ex. 200,000J/mole), recrystallization actually appears to occur at some definite minimum temp. 20

40 => this sensitivity makes it appear as though the metal has a fixed temp, below which it will not recrystallize. 21

41 8.12 The effect of Strain on Recrystallization Time for completion Of recrystallization (Amount of cold work) Rotary swaging 22

42 The slope is different: the temp. dependence of recrystallization (or Q R ) varies with the amount of deformation. Q R for recrystallization is a function of the amount of deformation. 23

43 8.13 The rate of Nucleation and The Rate of Nucleus Growth 1/ = Ae(-Q R /RT) reveals very little about the atomic mechanism because of the dual character of a nucleation and growth reaction. These two rates also determine the final grain size of a recrystallized metal. The introduction of the two parameters 24

44 Several equations have been derived and the theories on which these equations are based diverge to some extent. 25

45 8.13 The rate of Nucleation and The Rate of Nucleus Growth 1/ = Ae(-Q R /RT) reveals very little about the atomic mechanism because of the dual character of a nucleation and growth reaction. These two rates also determine the final grain size of a recrystallized metal. The introduction of the two parameters 1. N: the rate of nucleation (the number of nuclei that form per second in a cm 3 of unrecrystallized matrix) 1

46 Several equations have been derived and the theories on which these equations are based diverge to some extent. will not be discussed further. 2

47 8.14 Formation of Nuclei In recrystallization, an entirely new set of grains is formed. New crystals appear at points of high lattice strain energy, such as slip line intersections, deformation twin intersection, and in areas close to grain boundaries. nucleation occurs at points of strong lattice curvature The origin of a recrystallized grain is always a preexisting region that is highly misoriented in relation to the material surrounding it. This high degree of misorientation gives the region where the new grain originates the needed growth. 3

48 A number of models have been proposed to show how it is possible to form a small, strain-free volume that can grow out and consume the deformed matrix around it. These models are in general agreement on two points: 1. A region of a crystal can become a nucleus and grow only if its size exceeds some minimum value. 2. The formation of a nucleus is that it becomes surrounded (at least in part) by the equivalent of a high-angle grain boundary. 4

49 A single crystal lacks the sites along grain boundaries and along lines. For polycrystalline: three grains meet that are available for nucleation in a polycrystalline metal. One typical nucleation model for polycrystals: bulge mechanism: a difference in dislocation density exists across a grain boundary in a cold worked metal, then during annealing, the more perfect grain might migrate into the less perfect 5

50 If this bulge exceeds the critical nucleus size, both primary conditions for the formation of a nucleus would be satisfied. Less perfect grain (relatively strain-free volume of crystal) More perfect grain For polycrystals, nucleation will take place preferentially at highly energetic sites, such as grain-boundary triple points, original grain boundaries, and boundaries between deformation bands. 6

51 3 types of grain boundary nuclei marked by N by subgrain growth and/or grain boundary migration. The hexagonal networks are the subgrains. Subgrain growth Original GB GB migration the nucleus is formed by subgrain growth to the right of the original grain boundary. GB migration New GB Subgrain growth Subgrain 7

52 Other mechanisms for forming single crystals: 1. polygonization might be possible to produce a subgrain capable of growing out into the surrounding polygonized matrix. Subgrain coalescence Orientation different BCD and GHI straighten out 8

53 8.15 Driving Force for Recrystallization Coming from the stored energy of cold work. Polygonization is completed before recrystallization. The stored energy can be assumed to be confined to the dislocations in polygon walls. 9

54 8.16 Recrystallized Grain Size (another important factor) This is the crystal size immediately at the end of recrystallization before grain growth. Recrystallized grain size depends on the amount of deformation given to the specimens before annealing. The deformation step prior to recrystallization is critical to achieving uniform grain size. 10

55 When rolling is not performed efficiently, i.e. if thickness reduction is not large enough, the surface will undergo a lot of deformation while the core of the material will be only slightly deformed. C. Antoine, M. Foley, N. Dhanaraj Physical properties of niobium and specification of superconducting cavities August 25, 2006 Fermilab Technical Division Note TD

56 Temp effect The grain size is independent of the recrystallization temp. 3% recrystallize Amount of cold work => grain size => recrystallized grain size => recrystallization rate grain size control after primary deformations. 12

57 Dr. M. Medraj Mech. Eng. Dept. Concordia University Montreal, Canada The variation of recrystallization temperature with percent cold work for iron. For deformations less than the critical (about 5%), recrystallization will not occur. 13

58 The critical deformation, like the crystallization temp., is not a property of a metal, since its value varies with the type of deformation. Deformation occurs by easy glide, the critical deformation may exceed several hundred percent. The concept of a critical deformation is important The very large grain size associated with it is usually undesirable in metals that are to be further deformed. 14

59 If the grain size of a metal is very small, plastic deformation occurs without appreciable roughing of the surface. If the diameter of the average grain is large, cold working produces a roughened, objectionable surface. => orange-peel effect: a surface defect => Non-uniform deformation from grain to grain that produces the rough, orange peel condition. 15

60 Ratio of the rate of nucleation (N) to the rate of growth (G) 1. Small N/G (deformation ) => few nuclei, growth dominates, size 2. Large N/G (deformation ) => nucleation dominates, grain sizes bigger not much. 16

61 8.17 Other Variables in Recrystallization Factors: 1. temp. annealing: T, (slide 35 th ) but, did not affect grain size 2. amount of deformation (formation of nuclei) 3. purity (compositions) 4. initial grain size 8.18 Purity of the Metal Effect of purity of metal on the recrystallization => Purity, recrystallization rate 1

62 The effect is most significant at very small concentration of solute and the nature of the solute atoms. Conc. effect Solute effect Major reason: interaction of solute atom with grain boundary => when a foreign atom migrates to a grain boundary, both its elastic field, as well as that of the boundary, are lowered. 2

63 8.19 Initial Grain Size When a polycrystalline metal is cold worked, grain boundaries act to interrupt the slip processes that occur in the crystal. Recrystallization grain size => grain boundaries, the volume and uniformity of distorted metal (adjacent to the boundaries) 3

64 8.20 Grain Growth: Driving force for grain growth: reduction of the surface energy of grain boundary. Case of soap bubble: idea case of cellular growth. Because a number of complicated factors that influence the growth of metal crystals do not apply in the case of soap films First, consider a single spherical soap bubble => the gas enclosed by the soap film is always at a greater pressure than that on the outside of the bubble 4 because of the surface tensions in the soap film.

65 Gibb-Thompson equation: pressure difference p = 4 /R = 2*4 /(2R) = 8 /D : surface tension of one surface of the film R: the radius of the soap bubble D: diameter For case of p 1 and p 1* : net flux is coming out of the droplet Diffusion flux p 1 < P 1 * 5

66 The more general example: a soap froth Small, three- sided cells Concave toward the center => Shrink => disappear (time) A pressure difference exists across each curved wall 6

67 Smaller cells usually have fewest number of sides Cells with less than six sides have walls primarily concave toward their centers. => Both small and few sides: large curvature. => Accompanying high pressure differentials, diffusion rates, rates of wall migration. => unstable and tend to shrink in size. 120 o Shrink Stable grow 7

68 Typically, larger grains are easier to have sides more than 6. => grow at the expanses of small grain => grain growth or grain coarsening. 8 Boundary migrates New boundary between A and C. B and D each lose a side. A and C each gain a side.

69 8.21 Geometrical Coalescence: 2D Grains A and B encounter Separated by C Grain C is shrinking 9 sides rapid growth ab high angle grain boundary ab is with high surface energy ( G ) Grains A and B have nearly identical orientation; ab low angle grain boundary ab is with low surface energy ( G ) A single crystal Nine sides: possibility of continued rapid growth 9

70 Coalescence Orientation different BCD and GHI straighten out Recovery Recrystallization 10

71 Geometrical coalescence should have a strong effect on grain growth kinetics. On the other hand Geometrical coalescence may be an important phenomenon in a highly textured (strong preferred orientation). 11

72 8.22 Three-dimensional Changes in Grain Geometry Similar to slide 65 th, loss of 4 grain boundaries Similar to slide 67 th, loss of the line, gain a horizontal grain boundaries upper lower grain High energy GB upper low energy GB lower grain Inverse of B Geometrical coalescence Upper and lower approach each other, form a low energy grain boundary Inverse of D 12

73 8.23 The Grain Growth Law (still some discrepancies) Cell size: average diameter of the cells (soap froth) Grain size: mean diameter of an aggregate of grains (metals) Grain (cellular) growth: the growth of the average diameter of the aggregate. dd/dt = K c Assume c 1/D, K': constant; c: curvature of the cell walls D: mean diameter of the average cell dd dt K D DdD integration 2 Kdt D Kt c 13

74 D 2 = Kt + c Assume D = D 0 at t = 0 => D 2 D 02 = Kt If D 0 is samll => D 2 = Kt or D = (Kt) 1/2 = kt 1/2 (where k = K) 14

75 However, less is known about the mechanism by which atoms on one side of a grain boundary cross the boundary and join the crystal on the other side. convex Diffusion rate shrink concave Atoms tightly bound (more surrounded) If the diffusion of atoms across the boundary is an activated process => K = K 0 e -Q/kT (temp.) Q: empirical heat of activation => D 2 D 02 = K 0 te -Q/RT (grain growth form: time and temp.) D 2 t 15

76 Rearrange D 2 D 02 = K 0 te -Q/RT => (D 2 D 02 )/t = K 0 e -Q/RT => log[(d 2 D 02 )/t] = log(k 0 e -Q/RT ) = logk 0 + log(e -Q/RT ) => log[(d 2 D 02 )/t] 1/T; slope: Q/(2.3R) Slope: Q/(2.3R) to get Q log 16

77 Many of the experimental isothermal grain growth data correspond to empirical equations D = (Kt) 1/2 = kt 1/2 (where k = K) These discrepancies (not a simple activation law) come from 1. n is usually a function of temp (not a constant). 17

78 2 e e Q / RT Q / RT Review : the time required to recover a given fraction of the total yield point Q 1 1 ( ) R T T e Q: J/mole for Zn 1/ = Ae(-Q R /RT) Q R : activation energy for recrystallization 18

79 Grain growth factors: 1. Impurity atoms in solid solutions (8.24, grain boundary atmosphere) 2. Second phase inclusions (8.25) 3. Free surface effects (8.26) 4. Preferred orientation of the crystal structure (8.28) 19

80 8.24 Impurity Atoms in Solid Solution (1. grain boundary atmosphere) This interaction is analogous to the interaction between impurity atoms and dislocations (recrystallization, subgrain coalescence, slide 60 th ). If the size of a foreign atom and that of the parent crystal are different 20

81 We can conceive of grain boundary atmospheres just as we can dislocation atmospheres. Factors that affect the grain growth (grain boundary mobility) 1. Impurity atoms => interaction of solute atom with grain boundary 21

82 (impurity concentration) The approach of n to 0.5 is quicker for high T; assuming higher T (thermal vibrations) broke grain boundary solute atmosphere. The effect of solute in retarding grain growth varies with the element concerned. => due to different magnitudes of strain that various elements produce in the lattice. 22

83 8.25 Impurities in the Form of Inclusion (2.) Solute atoms not in solid solution are also capable of interacting with grain boundaries. Well known that impurity atoms in the form of second-phase or particles can inhibit grain growth in metals (8.27). Mechanical equilibrium (Surface tension force) 3D: circumference 2 r = 2 (rcos ) GB moves to the right 23

84 Grain boundary must pull itself through inclusions Grain boundary interacts with inclusions to induce surface tension stress ( ). The pull of the boundary on the particle f (nt/m) = ( sin )(2 rcos ) (nt/m) Maximum of f: = 45 o. => f = r. 24

85 An excellent example (inclusions dissolved) T grain size Conform with D = k(t) n Slope is the exponent n 25

86 Holes and pores have the same effect on grain boundary motion as second-phase inclusions. 26

87 8.26 The Free-Surface Effects (3.) Specimen geometry may play a part in controlling the rate of grain growth. Grain boundaries near any free surface of a metal tend to lie to the specimen surface, reducing the net curvature of the boundaries next to the surface. p = 4 /R = 2*4 /(2R) = 8 /D 1

88 Grain boundary grooves are more important in grain growth because they tend to anchor the ends of the grain boundaries, especially if the boundaries are nearly normal to the surface. A polished polycrystal has a flat surface. At room temperature, the surface remains flat for a long time. imechanica.org/files/l07%20%20gb%20grooving.pdf 2

89 Grain boundary grooves tend to anchor the ends of the G. B. / /2 b (free surface) fs sin(90 - /2) x 2 = b b = 2 fs cos( /2), b 3

90 To free the boundary from its groove requires work Total surface area, total surface energy The groove restrains the movement of the boundary. A boundary attached to its groove. Increase the total surface area, and total surface energy 4

91 8.27 Limiting grain size The specimen dimensions can influence the rate of grain growth when the average crystal size approaches the thickness of the specimen. 5

92 Consider a wire (with large grain) => boundaries cross the wire (on the free surface) => No curvature and can not migrate under action of surface tension forces => Further grain growth is then not possible. 6

93 A soap bubble has two spherical surfaces (inside and outside) with a thin layer of liquid in-between. Metal grain boundaries have only a single surface. 7

94 G.B. is no longer able to pull itself away from its inclusion, for a numbers of inclusions per unit area, the total restraining force is equal to. = 2 /R (from GB) = n s r (from the inclusions, slide 83 th ) n s : the numbers of inclusions per unit area Total volume: 2r*A Total # of particles: 2r*A*n v # of particles per unit area: n s = 2r*A*n v /A = 2r*n v 8

95 n s = 2r*A*n v /A = 2r*n v = 2 /R = n s r = (2rn v ) r ( (zeta) = n v *4/3 r 3 ) = 2r[ /(4/3 r 3 )] ( r ) = 6 /4r : the volume fraction of the second phase R: the radius of the average grain 9

96 8.28 Preferred Orientation (4.) By a preferred orientation, one signifies a nearly identical orientation in all the crystals of a given sample of metal. => generally observed grain growth rates are reduced Secondary Recrystallization (abnormal grain growth) (surface energy consideration) Growth inhibiting factors (1,2,3,4) for grain growth after primary recrystallization, do not control the growth of grains in secondary recrystallization. => A secondary recrystallization is often possible. 10

97 J. of Materials Science 8, 1-10 (1973). After completion of primary recrystallization, the whole polycrystal is relatively strain-free, and the grains, now in contact with one another, have approximately the same low dislocation density. At suitably higher temperatures, a new grain-growth process sets in. In these conditions the main driving force of the growth is the surface energy of large-angle grain boundaries; the boundary moves because the increase of the mean size of the grains reduces the total surface of the interfaces in the overall polycrystal volume, and therefore reduces the total boundary energy. The free energy of this process is relatively slight; the energy stored during cold-working is greater by one or two orders of magnitude even without very large deformations. The process by which grain size increases after primary recrystallization can occur according to two different patterns. (1) Normal (or continuous) grain growth, during which all the crystals grow in a uniform way; (2) abnormal (or discontinuous) grain growth, consisting in an exaggerated growth of only a few larger grains at the expense of the many smaller ones. This second kind of process, also known as secondary recrystallization, occurs in the presence of conditions which can restrain or inhibit normal grain growth and 11 transform its kinetics.

98 Difference between primary and secondary recrystallization: (a) primary: 1. new grain from deformed matrix (cold work) 2. strain free grain (b) secondary: 1. exaggerated grain growth, concave away grain from fine grain matrix.

99 13 sides Many sides (concave away) crystals consume the neighboring crystals. 13

100 8.30 Strain-induced boundary migration (lower strain energy) Larger distortion Surface energy may increase (surface area ) Lower distortion Normal grain growth: as a result of the surface energy stored in the grain boundaries (surface tension induced migration). 14

101 Strain-induced boundary movements differ from recrystallization. No new crystals are formed. One grain, the other. Interesting thing: Driving force: the reduction of the strain energy 15

102 Strain induced migration of boundaries only occurs after relatively small or moderate amounts of cold work. Too great a degree of deformation will bring about normal recrystallization Or induced where the normal grain growth has been inhibited by the size effects. 16

103 J. Appl. Phys. 21, 150 (1950) Grain boundary migration: reduction of the surface energy. A distinctive feature of this type of migration is that the boundaries move in the direction of their centers of curvature. The driving energy of grain growth is the excess free energy associated with the grain boundary surfaces. Grain growth tends to decrease this free energy, thus allowing the structure to approach equilibrium, by decreasing the grain boundary curvatures and the total grain boundary surface per unit volume. Grain boundary migration: reduction of the strain energy. In this case the moving boundaries separate strain hardened grains from annealed grains, and the direction of the movement is such that the volume of the annealed material increases with time at the expense of the cold worked material. The recrystallization involves the formation of nuclei of new strain free grains in the midst of the strain hardened material. During the growth of these new grains, their convex boundaries move in a direction opposite to their centers of curvature. 17