Gottstein G., Shvindlerman L.S. Grain Boundary Migration in Metals: Thermodynamics, Kinetics, Applications

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536 6 Applications

17a

13a

25a

Σ1

21a

25b

Σ1

21b

17b

Σ1

25a

25b

21b

25b

17b

sample: Sigma Relationships

levels: 2 4 8 16

scatter-width: 2°

fmax: 21

sections: 5° [001]

sym.

[010]

[100]

FIGURE 6.20

Location of low Σ orientation relationships in Rodrigues space.

6.2 Recrystallization annd Grain Growth 537

is mechanically stable. By heat treatment suﬃcient thermal energy is intro-

duced into the crystal to destabilize the dislocation arrangement. In principle,

there are two ways to remove the dislocations, or at least to reduce the stored

strain energy associated with the dislocations, namely recovery and recrystal-

lization. While recovery comprises all processes, which result in an annihilation

or rearrangement of dislocations in low energy dislocation structures, recrys-

tallization proceeds by generation and motion of grain boundaries, which are

capable of removing the swept-up dislocations. Therefore, while recovery will

locally rearrange and reduce the dislocation density, recrystallization will com-

pletely destroy the dislocation structure and rebuild a new strain-free crystal

arrangement. In the context of grain boundary migration, we shall focus on

recrystallization phenomena in the following.

Usually recrystallization proceeds from an abundance of recrystallization

nuclei and, therefore, the completely recrystallized material contains a high

density of grain boundaries. Owing to insuﬃcient conﬁguration entropy, grain

boundaries — like dislocations— also are not structural essentials of ther-

modynamic equilibrium and thus liable to be removed. As a consequence,

ﬁne-grained recrystallized microstructures undergo grain growth, driven by

the increase in grain boundary energy. Grain growth can proceed in a contin-

uous or discontinuous way. During continuous grain growth, some grains will

shrink while others will grow, but the average grain size will grow while the

grain size distribution remains self-similar, i.e. normalized with the average

grain size the grain size distribution will not change upon continuous grain

growth (Fig. 6.21a). In contrast, during discontinuous grain growth, few grains

grow at the expense of the other non-growing grains. The microstructural evo-

lution bears a resemblance to the process of recrystallization and, therefore, is

referred to as secondary recrystallization. During discontinuous grain growth

a bimodal grain size distribution will develop and, thus, the self-similarity of

the grain size distribution is not maintained in this process (Fig. 6.21b).

There is a confusing variety of terms for the various microstructural changes

during heat treatment of deformed materials. The typical recrystallization pro-

cess during heat treatment of a cold worked metal is referred to as static pri-

mary recrystallization or short recrystallization. Sometimes it is also referred

to as discontinuous recrystallization, since it proceeds locally and thus does

not aﬀect the entire volume concurrently. In contrast, continuous recrystalliza-

tion is not a real recrystallization process, but rather describes the phenomena

of very strong recovery, which also may result in the generation of large angle

grain boundaries. The term static recrystallization is used to distinguish the

two-step process, namely cold work and subsequent heat treatment, from the

recrystallization phenomena occurring concurrently with deformation at ele-

vated temperatures, which is referred to as dynamic recrystallization.

In contrast to the atomistic mechanisms of recrystallization the energetic

causes of recrystallization are relatively well understood. There is always a

driving force on a grain boundary, if — at a constant pressure and tempera-

ture — the displacement of the boundary reduces the Gibbs free energy G of

538 6 Applications

frequency of grains of

a certain size

ln D

m,u

ln D

(a) (b)

frequency of grains of

a certain size

frequency of grains of

a certain size

frequency of grains of

a certain size

ln D

m,u

ln D

m,u

ln D

(a) (b)

frequency of grains of

a certain size

frequency of grains of

a certain size

FIGURE 6.21

Time dependence of grain size distribution for normal (a) and abnormal (b)

grain growth (schematic).

the solid. The reduction of Gibbs free energy by displacement of a boundary

element of size d

A by the distance dx reads

dG = −Pd

Adx = −PdV (6.31)

where dV is the volume swept by the boundary during its migration. The

quantity

P = −

(6.32)

is referred to as driving force. It is a free energy gain per unit volume, which

equivalently can be understood as a force per unit area of the boundary, i.e.

as a pressure on the boundary. The driving force for primary recrystallization

is the strain energy of stored dislocations. If these dislocations are consumed

by the moving grain boundary, then the driving force is given by

P = ρE

ρμb

(6.33)

where E

is the energy per unit length of a dislocation (μ — shear modulus,

b — Burgers vector).

With a dislocation density ρ =10

−2

, μ =5·10

MPa, and b =2·10

−10

the driving force amounts to about P = 10 MPa, which has been conﬁrmed

by calorimetric measurements of the stored energy of cold work.

The driving force for grain growth is provided by the energy of the grain

boundaries. For discontinuous grain growth (secondary recrystallization) the

driving force reads

P =

3γ

(6.34a)

6.2 Recrystallization annd Grain Growth 539

whereitisassumedthattherecrystallized and thus consumed grains comprise

cubes of size d and γ is the grain boundary energy per unit area. For a re-

crystallized grain size of d =10

−4

m and a grain boundary energy γ =1J/m

one obtains P =0.03 MPa. Obviously, the driving force for secondary re-

crystallization is smaller by orders of magnitude than the driving force for

primary recrystallization. Therefore, grain growth phenomena proceed much

more slowly or at much higher temperatures compared to recrystallization.

For continuous grain growth an element of a grain boundary feels the driving

force for migration by the curvature of the boundary. This curvature is due to

the fact that grain boundary triple junctions have to comply with the force

equilibrium of surface tensions. For 3D crystal arrangements it is impossible

to establish both planar grain boundaries and equilibrium conditions at the

triple lines or quadruple junction where grain boundaries meet. Therefore,

polycrystals are always liable to grain growth and particular measures have

to be taken to suppress grain growth, for instance, to maintain a station-

ary microstructure in metallic parts during high temperature service, e.g. of

turbine blades in jet engines. The driving force for continuous grain growth

reads

P =

2γ

(6.34b)

where R is the radius of curvature. Since the radius of curvature R usually

exceeds the grain size by about an order of magnitude, the driving force for

continuous grain growth is about an order of magnitude smaller than for sec-

ondary recrystallization.

Reference is sometimes made to a process termed tertiary recrystallization.

This usually occurs in thin sheets, when grain growth has ceased due to groov-

ing of the grain boundaries exposed to the sheet surface. In such a case, the

diﬀerence in surface energy γ

of the various grains acts as a driving force on

the grain boundaries, and this driving force may exceed the driving force for

grain growth, if the surface energy diﬀerentials Δγ

are large and the sheet

thickness h is small, since the driving force reads (Fig. 6.22)

P =

2Δγ

(6.35)

Forinstance,forΔγ

=0.1J/m

and a sheet thickness h =10

−4

m the driving

force P =2·10

−3

MPa. Since the surface energy can be modiﬁed by adsorp-

tion, the process of tertiary recrystallization depends on the atmosphere, and

by changing atmospheres it is even possible to invert the direction of grain

boundary motion.

6.2.2 Nucleation of Recrystallization

As previously mentioned, the deformed microstructure is mechanically stable.

To destabilize the dislocation arrangement after deformation three criteria

540 6 Applications

FIGURE 6.22

Illustration of the driving force of tertiary recrystallization for γ

<γ

r > r

high angle

grain

boundary

pre-existing

nucleus

FIGURE 6.23

Schematic illustration of a recrystallization nucleus with growth potential in

a deformed structure.

6.2 Recrystallization annd Grain Growth 541

have to be met (instability criteria (Fig. 6.23)).

Thermodynamic Instability

A nucleus can only grow if the total free energy decreases during expansion

of the nucleus. For this a critical nucleus size r

has to be exceeded

2γ

4γ

ρμb

(6.36)

Since the nucleation rate due to thermal ﬂuctuations is much too small to

set oﬀ recrystallization, a supercritical nucleus must already be present in the

deformed microstructure. Usually, the deformation microstructure consists of

cellular arrangements, like dislocation cell structures or subgrain structures.

Thus, it is commonly assumed that there is a cell or subgrain of supercritical

size present in the microstructure to act as a viable recrystallization nucleus.

Mechanical Instability

Since the nucleus is embedded in a cellular arrangement of similar struc-

ture, there is not necessarily a net driving force on its boundaries for nucleus

growth; rather, it may also shrink or experience no net driving force at all

on its surface. An instability of the microstructure for recrystallization, there-

fore, requires that there is a net driving force for growth on at least part of

the surface of a nucleus. This can be achieved by an inhomogeneous dislo-

cation density distribution or a broad subgrain size distribution, which also

comprises large cells or subgrains. The latter can potentially be produced by

recovery.

Kinetic Instability

The most stringent requirement is, however, that the surface of the nucleus

must be a mobile grain boundary. As evident from Chapter 3, this requires

a large angle grain boundary. The generation of a mobile large angle grain

boundary in a deformed microstructure is the most diﬃcult step for nucle-

ation of recrystallization. Potential mechanisms may be nucleation at preex-

isting grain boundaries, discontinuous subgrain growth in orientation gradi-

ents, deformation inhomogeneities with large misorientations and generation

of annealing twins, in particular annealing twin chains.

The requirement for concurrent compliance with all three instability criteria

strongly prefers nucleation in certain locations of the deformed microstructure,

in particular in deformation inhomogeneities (Fig. 6.24a) like shear bands or

microbands, or at preexisting large angle grain boundaries (Fig. 6.24b). An

exception is the formation of annealing twins, which by continued twinning

and thus generation of a twin chain (twin family) may produce a mobile large

angle grain boundary (Fig. 6.25). So far, observations have only conﬁrmed

that twinning proceeds from a moving grain boundary, and thus already re-

quires the formation of a viable recrystallization nucleus.

542 6 Applications

FIGURE 6.24

(a) TEM image of a recrystallized nucleus that developed near the edge of a

shear band and is growing into the deformed structure; (b) nucleation at the

grain edges of zone annealed aluminum.

FIGURE 6.25

Micrograph and {111} pole ﬁgure of a copper single-crystal after dynamic

recrystallization during a tensile test at 1103 K. ( −−−ﬁrst-; −·−second-;

−··−third-generation twinning.)

6.2 Recrystallization annd Grain Growth 543

2468100

0.2

0.4

0.6

0.8

1.0

fraction recrystallized x

time [10

5.1%

350°C

2468100

0.2

0.4

0.6

0.8

1.0

fraction recrystallized x

time [10

s]time [10

5.1%

350°C

5.1%

350°C

FIGURE 6.26

Recrystallized volume fraction as a function of annealing time following 5.1%

tensile deformation (after [581]).

6.2.3 Kinetics of Primary Recrystallization

The kinetics of recrystallization, i.e. the change of recrystallized volume frac-

tion X with annealing time t (Fig. 6.26) is determined by thermal activation

of the recrystallization mechanisms nucleation and nucleus growth. From nu-

merous experimental investigations, it is evident that the recrystallization

kinetics can be described by the Avrami-Johnson-Mehl-Kolmogorov equation

[578]–[580]

X =1−exp



−







(6.37)

In this equation q is the Avrami exponent and t

a characteristic time for

recrystallization, i.e. when 63.2% of the volume is recrystallized. In the case

of a constant and homogeneous nucleation rate

N and isotropic growth rate

v the above equation can be derived as

X(t)=1−exp



−



(6.38)

Comparing Eq. (6.38) with (6.37) the characteristic time for recrystallization

and the recrystallized grain size d





−1/4

(6.39)

d =2vt

∼





1/4

(6.40)

544 6 Applications

Evidently, the recrystallized grain size is the result of a competition between

growth rate and nucleation rate. In contrast, the recrystallization time is a

joint eﬀect of nucleation and growth, although very strongly dominated by

the growth rate. Apparently, the grain boundary migration rate, and thus

the grain boundary mobility, governs the kinetics of recrystallization and the

characteristics of the recrystallized microstructure.

6.2.4 Recrystallization Texture

6.2.4.1 Origin of Textures

The dominant deformation mechanism in metals is crystallographic slip by dis-

location glide. Crystallographic slip is a pure shear deformation and, therefore,

accompanied by rigid body rotation, which is the cause for the development of

deformation textures during cold work. Depending on material, temperature,

strain rate and, of course, strain, a speciﬁc deformation texture will be formed,

for instance, during rolling, which is the most common industrial forming pro-

cess. Interestingly, for a given rolling texture a deﬁned recrystallization texture

develops upon annealing

. As an example, a conspicuously pronounced Cube

texture is found upon recrystallization of rolled copper (Fig. 6.27a,b). The tex-

ture is very sharp, almost like a single crystal. Equivalently, the typical brass

recrystallization texture develops from the brass rolling texture (Figs. 6.27c,d).

The respective orientations and their volume fractions are listed in Table 6.3.

There is also a speciﬁc texture development in bcc materials during annealing

of rolled sheet, for instance in ferritic steels (Fig. 6.28).

Because of the importance of recrystallization textures for commercial ap-

plications, such as industrial forming processes or electrical steels, there have

been considerable eﬀorts to account for the formation of a recrystallization

texture from a given deformation texture. Since the process of recrystalliza-

tion comprises the stages nucleation and growth, theories were developed that

either attributed the recrystallization texture to the preference of special ori-

entations to be nucleated or the preference of special misorientations to grow,

i.e. to high mobility grain boundaries [582]–[584]. Extensive investigations into

this issue in the past 30 years have substantiated that there are many more

nucleus orientations than recrystallization texture components and also that

there is no growth competition among randomly disoriented grains. Rather,

it is now common belief that recrystallization textures are formed by growth

competition for a limited spectrum of nucleus orientations. Because of the

extreme sensitivity of the recrystallization kinetics on the growth rate, as evi-

dent from Eq. (6.38), essentially only the maximum growth rate misorientation

dominates the recrystallization texture development. Therefore, considerable

For ideal orientations of rolling and recrystallization textures in cubic crystals see Figs. 6.12

and 6.13 and Tables 6.1 and 6.2.

6.2 Recrystallization annd Grain Growth 545

eﬀorts were dedicated to a determination of the maximum growth rate mis-

orientation by so-called growth selection experiments [585].

TABLE 6.3

Pure Cu: Rolling Texture

Φ ϕ

f (g

) M

{hkl}uvw OR Name

Pure Cu: Rolling Texture

35 45 0 15.1 29.5 7.9 011 211 Brass-Def

62 33.5 63 18.5 42.1 7.4 123 634 S

90 32 45 16.9 25.5 8.4 112 111 Cu

90 80 45 5.5 2.8 6.7 255 511 Twin to Cu

0 21 0 4.1 1.8 5.6 025 100 Tilted Cube

Pure Cu: Recrystallization Texture

0 0 0 90.6 52.6 7.1 001 100 Cube

63 48 63 22.9 20.0 6.5 122 212 1

Twin to Cube

16.5 26 10.0 6.8 9.5 8.7 148 841 2

Twin to Cube

76 29 74 6.6 9.5 9.3 148 747 2

Twin to Cube

90 35 0 11.5 2.7 7.8 035 053 Tilted Cube

45 37 45 6.0 3.9 6.9 447 184 2

Twin to Cube

Cu-22% Zn (Brass): Rolling Texture

32 45 0 13.8 65.2 13.2 011 211 Brass-Def

0 45 0 5.5 17.3 14.8 011 100 Goss

90 66 45 3.4 10.5 12.4 233 311 {223}

0 51 45 2.7 7.0 10.6 111 110 {111}

Cu-22% Zn (Brass): Recrystallization Texture

82 30 34 9.0 63.2 12.0 236 385 Brass Rex

90 33 0 4.4 8.9 10.4 035 053 Tilted Cube

0 20 0 2.7 8.1 11.6 025 100 Tilted Cube

45 38 68 2.5 18.5 12.1 134 132 –

0 0 0 1.2 0.3 7.8 001 100 Cube

Basic types of rolling and recrystallization textures in fcc metals:

— volume fraction; f (g

)—valueofODFatg

;

— scattering amplitude; OR — name of orientation [590].