**2. Theories for evolution of recrystallization textures**

Rex occurs by nucleation and growth. Therefore, the evolution of the Rex texture must be controlled by nucleation and growth. In the oriented nucleation theory (ON), the preferred activation of a special nucleus determines the final Rex texture [1]. In the oriented growth theory (OG), the only grains having a special relationship to the deformed matrix can pref‐ erably grow [2]. Recent computer simulation studies tend to advocate ON theory [3]. This comes from the presumption that the growth of nuclei is predominated by a difference in

© 2013 Lee and Han; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

energy between the nucleus and the matrix, or the driving force. In addition to this, the weakness of the conventional OG theory is in much reliance on the grain boundary mobility.

ment controlled system. When a stress-free specimen *S*0 is elastically elongated by ∆*L* by force *FA* (Figure 2a), the elongated specimen *SF* has an elastic strain energy represented by triangle *OAC* (Figure 2b). When V in *SF* is replaced by a stress-free volume V, *SR* having the stress free V has the strain energy of *OBC* (Figure 2b. ) Transformation from the *SF* state to the *SR* state results in a strain-energy-release represented by *OAB* (Figure 2b). The strain-energy-release can be maximized when the *SF* and *SR* states have the maximum and minimum strain energies, respectively. In this case, AMSD is the axial direction of *SF*, and the *SR* state has the minimum energy when MYMD of the stress-free V is along the axial direction that is AMSD. In summary, the strain energy release is maximized when AMSD in the high dislocation density matrix is along MYMD of the stress free crystal, or nu‐ cleus. That is, when a volume of *V* in the stress field is replaced by a stress-free single crystal of the volume *V*, the strain energy release of the system occurs. The strain energy release can change depending on the orientation of the stress-free crystal. The strain ener‐ gy release is maximized when AMSD in the high energy matrix is along MYMD of the stress-free crystal. The stress-free grains formed in the early stage are referred to as nu‐ clei, if they can grow. The orientation of a nucleus is determined such that its strain ener‐

**Figure 1.** (a) Schematic dislocation array after recovery, where horizontal arrays give rise to long-range stress field, and vertical arrays give rise to short-range stress field [7]. Principal stress distributions around parallel edge disloca‐ tions calculated based on (b) 100 linearly arrayed dislocations with dislocation spacing of 10**b**, and (c) low energy ar‐

*A*

*B*

*C*

matrix

c

MYMD AMSD //

Recrystallization Textures of Metals and Alloys

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5

high dislocation density

recrystallized grain

b

*FA* energy strain release

*L*

**Figure 2.** Displacement controlled uniaxial specimen for explaining strain-energy-release being maximized when

gy release per unit volume during Rex becomes maximized.

ray of 100 x 100 dislocations. **b** is Burgers vector and G is shear modulus [8].

*F*

*FA*

AMSD in high dislocation density matrix is along MYMD in recrystallized grain.

*O*

a

*<sup>L</sup> <sup>O</sup> <sup>S</sup> F S*

*<sup>R</sup> V S*

One of the present authors (Lee) advanced a theory for the evolution of Rex textures [4] and elaborated later [5,6]. In the theory, the Rex texture is determined such that the absolute maximum stress direction (AMSD) due to dislocation array formed during fabrication and subsequent recovery is parallel to the minimum Young's modulus direction (MYMD) in recrystallized (Rexed) grains and other conditions are met, whereby the strain energy release can be maximized. In the strain-energy-release-maximization theory (SERM), elastic anisotro‐ py is importantly taken into account.

In what follows, SERM is briefly described. Rex occurs to reduce the energy stored during fabri‐ cation by a nucleation and growth process. The stored energy may include energies due to va‐ cancies, dislocations, grain boundaries, surface, etc. The energy is not directional, but the texture is directional. No matter how high the energy may be, the defects cannot directly be re‐ lated to the Rex texture, unless they give rise to some anisotropic characteristics. An effect of ani‐ sotropy of free surface energy due to differences in lattice surface energies can be neglected except in the case where the grain size is larger than the specimen thickness in vacuum or an in‐ ert atmosphere. Differences in the mobility and/or energy of grain boundaries must be impor‐ tant factors to consider in the texture change during grain growth. Vacancies do not seem to have an important effect on the Rex texture due to their relatively isotropic characteristics. The most important driving force for Rex (nucleation and growth) is known to be the stored energy due to dislocations. The dislocation density may be different from grain to grain. Even in a grain the dislocation density is not homogeneous. Grains with low dislocation densities can grow at the expanse of grains with high dislocation densities. This may be true for slightly deformed metals as in case of strain annealing. However, the differences in dislocation density and orien‐ tation between grains decrease with increasing deformation. Considering the fact that strong deformation textures give rise to strong Rex textures, the dislocation density difference cannot be a dominant factor for the evolution of Rex textures. Dislocations cannot be related to the Rex texture, unless they give rise to anisotropic characteristics.

The dislocation array in fabricated materials looks very complicated. Dislocations generat‐ ed during plastic deformation, deposition, etc., can be of edge, screw, and mixed types. Their Burgers vectors can be determined by deformation mode and texture, and their ar‐ ray can be approximated by a stable or low energy arrangement of edge dislocations af‐ ter recovery. Figure 1 shows a schematic dislocation array after recovery and principal stress distributions around stable and low energy configurations of edge dislocations, which were calculated using superposition of the stress fields around isolated disloca‐ tions, or, more specifically, were obtained by a summation of the components of stress field of the individual dislocations sited in the array. It can be seen that AMSD is along the Burgers vector of dislocations that are responsible for the long-range stress field. The volume of crystal changes little after heavy deformation because contraction in the com‐ pressive field and expansion in the tensile fields around dislocations generated during de‐ formation compensate each other. That is, this process takes place in a displacement controlled system. The uniaxial specimen in Figure 2 makes an example of the displace‐ ment controlled system. When a stress-free specimen *S*0 is elastically elongated by ∆*L* by force *FA* (Figure 2a), the elongated specimen *SF* has an elastic strain energy represented by triangle *OAC* (Figure 2b). When V in *SF* is replaced by a stress-free volume V, *SR* having the stress free V has the strain energy of *OBC* (Figure 2b. ) Transformation from the *SF* state to the *SR* state results in a strain-energy-release represented by *OAB* (Figure 2b). The strain-energy-release can be maximized when the *SF* and *SR* states have the maximum and minimum strain energies, respectively. In this case, AMSD is the axial direction of *SF*, and the *SR* state has the minimum energy when MYMD of the stress-free V is along the axial direction that is AMSD. In summary, the strain energy release is maximized when AMSD in the high dislocation density matrix is along MYMD of the stress free crystal, or nu‐ cleus. That is, when a volume of *V* in the stress field is replaced by a stress-free single crystal of the volume *V*, the strain energy release of the system occurs. The strain energy release can change depending on the orientation of the stress-free crystal. The strain ener‐ gy release is maximized when AMSD in the high energy matrix is along MYMD of the stress-free crystal. The stress-free grains formed in the early stage are referred to as nu‐ clei, if they can grow. The orientation of a nucleus is determined such that its strain ener‐ gy release per unit volume during Rex becomes maximized.

energy between the nucleus and the matrix, or the driving force. In addition to this, the weakness of the conventional OG theory is in much reliance on the grain boundary mobility.

One of the present authors (Lee) advanced a theory for the evolution of Rex textures [4] and elaborated later [5,6]. In the theory, the Rex texture is determined such that the absolute maximum stress direction (AMSD) due to dislocation array formed during fabrication and subsequent recovery is parallel to the minimum Young's modulus direction (MYMD) in recrystallized (Rexed) grains and other conditions are met, whereby the strain energy release can be maximized. In the strain-energy-release-maximization theory (SERM), elastic anisotro‐

In what follows, SERM is briefly described. Rex occurs to reduce the energy stored during fabri‐ cation by a nucleation and growth process. The stored energy may include energies due to va‐ cancies, dislocations, grain boundaries, surface, etc. The energy is not directional, but the texture is directional. No matter how high the energy may be, the defects cannot directly be re‐ lated to the Rex texture, unless they give rise to some anisotropic characteristics. An effect of ani‐ sotropy of free surface energy due to differences in lattice surface energies can be neglected except in the case where the grain size is larger than the specimen thickness in vacuum or an in‐ ert atmosphere. Differences in the mobility and/or energy of grain boundaries must be impor‐ tant factors to consider in the texture change during grain growth. Vacancies do not seem to have an important effect on the Rex texture due to their relatively isotropic characteristics. The most important driving force for Rex (nucleation and growth) is known to be the stored energy due to dislocations. The dislocation density may be different from grain to grain. Even in a grain the dislocation density is not homogeneous. Grains with low dislocation densities can grow at the expanse of grains with high dislocation densities. This may be true for slightly deformed metals as in case of strain annealing. However, the differences in dislocation density and orien‐ tation between grains decrease with increasing deformation. Considering the fact that strong deformation textures give rise to strong Rex textures, the dislocation density difference cannot be a dominant factor for the evolution of Rex textures. Dislocations cannot be related to the Rex

The dislocation array in fabricated materials looks very complicated. Dislocations generat‐ ed during plastic deformation, deposition, etc., can be of edge, screw, and mixed types. Their Burgers vectors can be determined by deformation mode and texture, and their ar‐ ray can be approximated by a stable or low energy arrangement of edge dislocations af‐ ter recovery. Figure 1 shows a schematic dislocation array after recovery and principal stress distributions around stable and low energy configurations of edge dislocations, which were calculated using superposition of the stress fields around isolated disloca‐ tions, or, more specifically, were obtained by a summation of the components of stress field of the individual dislocations sited in the array. It can be seen that AMSD is along the Burgers vector of dislocations that are responsible for the long-range stress field. The volume of crystal changes little after heavy deformation because contraction in the com‐ pressive field and expansion in the tensile fields around dislocations generated during de‐ formation compensate each other. That is, this process takes place in a displacement controlled system. The uniaxial specimen in Figure 2 makes an example of the displace‐

py is importantly taken into account.

4 Recent Developments in the Study of Recrystallization

texture, unless they give rise to anisotropic characteristics.

**Figure 1.** (a) Schematic dislocation array after recovery, where horizontal arrays give rise to long-range stress field, and vertical arrays give rise to short-range stress field [7]. Principal stress distributions around parallel edge disloca‐ tions calculated based on (b) 100 linearly arrayed dislocations with dislocation spacing of 10**b**, and (c) low energy ar‐ ray of 100 x 100 dislocations. **b** is Burgers vector and G is shear modulus [8].

**Figure 2.** Displacement controlled uniaxial specimen for explaining strain-energy-release being maximized when AMSD in high dislocation density matrix is along MYMD in recrystallized grain.

S2, and the displacement Δ*x*3 along the *x*3 axis at any point P with coordinates (*x*1,*x*2,*x*3) is considered. If shear strains *γ*(1) and *γ*(2) occur on the slip system 1 (the slip plane S1 and the slip direction *x*3) and the slip system 2 (the slip plane S2 and the slip direction *x*3), respectively, then

> 

Because *α* > *β* and (*γ*(1) + *γ*(2)) > (*γ*(2) - *γ*(1)), the second term of the right hand side is negligible compared with the first term. It follows from OP cos*β* = *x*2 that Δ*x*<sup>3</sup> ≈ (*γ*(1) + *γ*(2)) *x*<sup>2</sup> sin*α*. Therefore, the displacement Δ*x*3 is linear with the *x*2 coordinate, and the deformation is equivalent to single slip in the *x*3 direction on the (*γ*(1)S1 + *γ*(2)S2) plane. The apparent shear strain *γa* is

(1 2 ) ( )

3 2 / ) sin( *<sup>a</sup>*

on the slip systems *i* are

( ) ( ) sin *i i*

For bcc metals, sin*α* = 0.5 (e.g. a duplex slip of (101)[1 1-1] and (011)[1 1-1]) and hence

( ) ( ) ( ) bcc 0.5 *<sup>i</sup> <sup>i</sup>*

For fcc metals, sin*α* = 0.577 (e.g. a duplex slip of (-1 1-1)[110] and (1-1-1)[110]) and hence

( ) ( ) ( ) fcc 0.577 *i i*

The activity of each slip direction is linearly proportional to the dislocation density *ρ* on the cor‐ responding slip system, which is roughly proportional to the shear strain on the slip system. Ex‐ perimental results on the relation between shear strain *γ* and *ρ* are available for Cu and Al [9].

If a crystal is plastically deformed by *δε* (often about 0.01), then we can calculate active slip

strain rate with respect to strain of specimen, *dγ*(*i*) */dε*. During this deformation, the crystal can rotate, and active slip systems and shear strains on them change during *δε*. When a crystal

 

> 

*a*

*a* 

*a* 

 


 

 » *x x* (4)

 

PN PN (1)

 

(5)

(6)

(7)

on them using a crystal plasticity model, resulting in the shear

OP sin cos - ( OP cos) sin (3)

(2)

Recrystallization Textures of Metals and Alloys

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7

(1 2 ) ( ) <sup>312</sup> *x*

where PN1 and PN2 are normal to the planes S1 and S2, respectively. Therefore,

1 2 PN = OP sin and PN = OP sin ( ) 

(1 2 ) ( ) (2 1 ) ( )

 

where OP, *α*, and *β* are defined in Figure 4. Therefore,

 

(*i*)

<sup>3</sup> *x* ( ) 

The apparent shear strains *γ<sup>a</sup>*

systems *i* and shear strains *γ*(*i*)

**Figure 3.** AMSD for active slip systems *i* whose Burgers vectors are **b**(i) and activities are γ(i).

**Figure 4.** Schematic of two slip planes S1 and S2 that share common slip direction along *x*3 axis.

We first calculate AMSD in an fcc crystal deformed by a duplex slip of (111)[-101] and (111) [-110] that are equally active. The duplex slip can be taken as a single slip of (111)[-211], which is obtained by the sum of the two slip directions. In this case, the maximum stress direction is [-211]. However, some complication can occur. One slip system has two opposite directions. The maximum stress direction for the (111)[-101] slip system represents the [-101] direction and its opposite direction, [1 0-1]. The maximum stress direction for the (111)[-110] slip system represents the [-110] and [1-1 0] directions. Therefore, there are four possible combinations to calculate the maximum stress direction, [-101] + [-110] = [-211], [-101] + [1-1 0] = [0-1 1], [1 0-1] + [-110] = [0 1-1], and [1 0-1] + [1-1 0] = [2-1-1], among which [-211]//[2-1-1] and [0-1 1]//[0 1-1]. The correct combinations are such that two directions make an acute angle. If the two slip systems are not equally active, the activity of each slip system should be taken into account. If the (111)[-101] slip system is two times more active than the (111)[-110] system, the maximum stress direction becomes 2[-101] + [-110] = [-312]. This can be generalized to multiple slip. For multiple slip, AMSD is calculated by the sum of active slip directions of the same sense and their activities, as shown in Figure 3. It is convenient to choose slip directions so that they can be at acute angles with the highest strain direction of the specimen, e.g., RD in rolled sheets, the axial direction in drawn wires, etc.

When two slip systems share the same slip direction, their contributions to AMSD are reduced by 0.5 for bcc metals and 0.577 for fcc metals as follows. Figure 4 shows two slip planes, S1 and S2, intersecting along the common slip direction, the *x*3 axis; the *x*2 axis bisects the angle between the poles of these planes. The loading direction lies within the quadrant drawn between S1 and S2, and the displacement Δ*x*3 along the *x*3 axis at any point P with coordinates (*x*1,*x*2,*x*3) is considered. If shear strains *γ*(1) and *γ*(2) occur on the slip system 1 (the slip plane S1 and the slip direction *x*3) and the slip system 2 (the slip plane S2 and the slip direction *x*3), respectively, then

$$
\Delta \mathbf{x}\_3 = \boldsymbol{\gamma}^{(1)} \cdot \mathbf{PN}\_1 + \boldsymbol{\gamma}^{(2)} \cdot \mathbf{PN}\_2 \tag{1}
$$

where PN1 and PN2 are normal to the planes S1 and S2, respectively. Therefore,

$$\text{PN}\_1 \text{=OP} \sin(a - \beta) \\ \text{and} \quad \text{PN}\_2 \text{=OP} \sin(a + \beta) \tag{2}$$

where OP, *α*, and *β* are defined in Figure 4. Therefore,

( AMSD//))2()2()1()1( )1( **bb** 

)1( *xxx* <sup>333</sup>

N1

We first calculate AMSD in an fcc crystal deformed by a duplex slip of (111)[-101] and (111) [-110] that are equally active. The duplex slip can be taken as a single slip of (111)[-211], which is obtained by the sum of the two slip directions. In this case, the maximum stress direction is [-211]. However, some complication can occur. One slip system has two opposite directions. The maximum stress direction for the (111)[-101] slip system represents the [-101] direction and its opposite direction, [1 0-1]. The maximum stress direction for the (111)[-110] slip system represents the [-110] and [1-1 0] directions. Therefore, there are four possible combinations to calculate the maximum stress direction, [-101] + [-110] = [-211], [-101] + [1-1 0] = [0-1 1], [1 0-1] + [-110] = [0 1-1], and [1 0-1] + [1-1 0] = [2-1-1], among which [-211]//[2-1-1] and [0-1 1]//[0 1-1]. The correct combinations are such that two directions make an acute angle. If the two slip systems are not equally active, the activity of each slip system should be taken into account. If the (111)[-101] slip system is two times more active than the (111)[-110] system, the maximum stress direction becomes 2[-101] + [-110] = [-312]. This can be generalized to multiple slip. For multiple slip, AMSD is calculated by the sum of active slip directions of the same sense and their activities, as shown in Figure 3. It is convenient to choose slip directions so that they can be at acute angles with the highest strain direction of the specimen, e.g., RD in rolled sheets,

When two slip systems share the same slip direction, their contributions to AMSD are reduced by 0.5 for bcc metals and 0.577 for fcc metals as follows. Figure 4 shows two slip planes, S1 and S2, intersecting along the common slip direction, the *x*3 axis; the *x*2 axis bisects the angle between the poles of these planes. The loading direction lies within the quadrant drawn between S1 and

)1( 

P

 PN PN

N2

)2( 

P

<sup>3</sup> *x* <sup>3</sup> *<sup>x</sup>*

2 )2( 1

)2()2( **b** 

**b**

**Figure 3.** AMSD for active slip systems *i* whose Burgers vectors are **b**(i) and activities are γ(i).

N2

S2

**Figure 4.** Schematic of two slip planes S1 and S2 that share common slip direction along *x*3 axis.

1 *x*

)2( **b**

 

3 *x*

O

N1

<sup>3</sup> *x* )( <sup>321</sup> *xxx*

2 *x*

P

6 Recent Developments in the Study of Recrystallization

S1

the axial direction in drawn wires, etc.

)1()1( **b** 

$$\Delta \mathbf{x}\_3 = \begin{pmatrix} \boldsymbol{\gamma}^{(1)} \ + \boldsymbol{\gamma}^{(2)} \end{pmatrix} \text{OP} \sin a \cos \boldsymbol{\beta} + \begin{pmatrix} \boldsymbol{\gamma}^{(2)} \ -\boldsymbol{\gamma}^{(1)} \end{pmatrix} \text{OP} \cos a \sin \boldsymbol{\beta} \tag{3}$$

Because *α* > *β* and (*γ*(1) + *γ*(2)) > (*γ*(2) - *γ*(1)), the second term of the right hand side is negligible compared with the first term. It follows from OP cos*β* = *x*2 that Δ*x*<sup>3</sup> ≈ (*γ*(1) + *γ*(2)) *x*<sup>2</sup> sin*α*. Therefore, the displacement Δ*x*3 is linear with the *x*2 coordinate, and the deformation is equivalent to single slip in the *x*3 direction on the (*γ*(1)S1 + *γ*(2)S2) plane. The apparent shear strain *γa* is

$$\text{If } \mathbf{y}\_a = \Delta \mathbf{x}\_3 / \mathbf{x}\_2 \approx (\mathbf{y}^{\{1\}} + \mathbf{y}^{\{2\}}) \sin a \tag{4}$$

The apparent shear strains *γ<sup>a</sup>* (*i*) on the slip systems *i* are

$$
\gamma\_a^{(i)} = \gamma^{(i)} \sin a \tag{5}
$$

For bcc metals, sin*α* = 0.5 (e.g. a duplex slip of (101)[1 1-1] and (011)[1 1-1]) and hence

$$
\gamma\_a^{(i)} \text{(bcc)} = 0.5 \gamma^{(i)} \tag{6}
$$

For fcc metals, sin*α* = 0.577 (e.g. a duplex slip of (-1 1-1)[110] and (1-1-1)[110]) and hence

$$
\gamma\_a^{(i)} \text{(fccc)} = 0.577 \,\gamma^{(i)} \tag{7}
$$

The activity of each slip direction is linearly proportional to the dislocation density *ρ* on the cor‐ responding slip system, which is roughly proportional to the shear strain on the slip system. Ex‐ perimental results on the relation between shear strain *γ* and *ρ* are available for Cu and Al [9].

If a crystal is plastically deformed by *δε* (often about 0.01), then we can calculate active slip systems *i* and shear strains *γ*(*i*) on them using a crystal plasticity model, resulting in the shear strain rate with respect to strain of specimen, *dγ*(*i*) */dε*. During this deformation, the crystal can rotate, and active slip systems and shear strains on them change during *δε*. When a crystal rotates during deformation, the absolute value of shear strain rates |*dγ*(*i*) */dε*| on slip systems *i* can vary with strain *ε* of specimen. For a strain up to *ε* = *e*, the contribution of each slip system to AMSD is proportional to

$$\gamma^{(i)} = \int\_0^\varepsilon \left| d\gamma^{(i)} \right> \, d\varepsilon \Big| \, d\varepsilon \tag{8}$$

123 123 123 123 123 123

It should be mentioned that *a* is set to be along [100], but *b* is along [010] or [001] depending on physical situations and c is consequently along [001] or [010]. The Rex texture can often be obtained without resorting to the above process because the AMSD//MYMD condition is so

The 1st priority: When AMSD is cristallographically the same as MYMD, No texture changes

The 2nd priority: When AMSD crystallographically differs from MYMD, the Rex texture is determined such that AMSD in the matrix is parallel to MYMD in the Rexed grain, with one common axis of rotation between the deformed and Rexed states. The common axis can be ND, TD, or other direction (e.g. <110> for bcc metals). This may be related to minimum atomic movement at the AMSD//MYMD constraints. However, we do not know the exact physical

The 3rd priority: When the first two conditions are not met, the method explained to obtain

When the density of dislocations in electrodeposits and vapor deposits is high, the deposits undergo Rex when annealed. AMSD in the deposits can be determined by their textures. The density of dislocations whose Burgers vectors are directed away from the growth direction (GD)of deposits was supposed to be higher than when the Burgers vector is nearly parallel to GD because dislocations whose Burgers vector is close to GD are easy to glide out from the deposits by the image force during their growth [11]. This was experimentally proved in a Cu electrodeposit with the <111> orientation [12]. Therefore, AMSDs are along the Burgers vectors

Lee et al. found that the <100>, <111>, and <110> textures (inverse pole figures: IPFs) of Cu electrodeposits which were obtained from Cu sulfate and Cu fluoborate baths [13,14], and a cyanide bath [15] changed to the <100>, <100>, and <√310> textures, respectively, after Rex as shown in Figure 7. The texture fraction (TF) of the (*hkl*) reflection plane is defined as follows:

> o o

*hkl hkl* åé ù ë û (10)

I( )/I ( ) TF( ) I( )/I ( ) *hkl hkl hkl*

(9)

9

Recrystallization Textures of Metals and Alloys

http://dx.doi.org/10.5772/54123

*h aaa h u aaa u k bbb k v bbb v l ccc l w ccc w*

æ ö æ öæ ö æ ö æ öæ ö ç ÷ ç ÷ç ÷ ç ÷ ç ÷ç ÷

è ø è øè ø è ø è øè ø

dominant that the Rex texture can be obtained by the following priority order.

*r r r r r r*

after Rex [10].

picture of this.

Eq. 9 is used.

nearly normal to GD.

**3. Electrodeposits and vapor-deposits**

**3.1. Copper, nickel, and silver electrodeposits**

The above equation is illustrated in Figure 5. If a deformation texture is stable, the shear strain rates on the slip systems are independent of deformation.

So far methods of obtaining AMSD have been discussed. This is good enough for prediction of fiber textures. However, the stress states around dislocation arrays are not uniaxial but triaxial. Unfortunately we do not know the stress fields of individual dislocations in real crystals, but know Burgers vectors. Therefore, AMSD obtained above applies to real crystals. Any stress state has three principal stresses and hence three principal stress directions which are perpendicular to each other. Once we know the three principal stress directions, the Rex textures are determined such that the three directions in the deformed matrix are parallel to three <100> directions in the Rexed grain, when MYMDs are <100>. In figure 6, let the unit vectors of **A**, **B**, and **C** be *a* [*a*<sup>1</sup> *a*<sup>2</sup> *a*3], *b* [*b*<sup>1</sup> *b*<sup>2</sup> *b*3], and *c* [*c*<sup>1</sup> *c*<sup>2</sup> *c*3], where *ai* are direction cosines of the unit vector *a* referred to the crystal coordinate system. AMSD is one of three principal stress directions. Two other principal stresses are obtained as explained in Figure 6.

**Figure 5.** Calculation of γ(*i*) for crystal rotation during deformation up to ε = *e.*

**Figure 6.** Relationship between three principal stress directions **A**, **B**, and **C**.

If the unit vectors *a*, *b*, and *c* are set to be along [100], [010], and [001] after Rex, components of the unit vectors are direction cosines relating the deformed and Rexed crystal coordinate systems, when MYMDs are <100>. That is, the (*hkl*)[*uvw*] deformation orientation is calculated to transform to the (*hr kr lr*)[*ur vr wr*] Rex orientation using the following equation.

$$
\begin{pmatrix} h\_r \\ k\_r \\ l\_r \end{pmatrix} = \begin{pmatrix} a\_1 & a\_2 & a\_3 \\ b\_1 & b\_2 & b\_3 \\ c\_1 & c\_2 & c\_3 \end{pmatrix} \begin{pmatrix} h \\ k \\ l \end{pmatrix} \begin{pmatrix} u\_r \\ v\_r \\ w\_r \end{pmatrix} = \begin{pmatrix} a\_1 & a\_2 & a\_3 \\ b\_1 & b\_2 & b\_3 \\ c\_1 & c\_2 & c\_3 \end{pmatrix} \begin{pmatrix} u \\ v \\ w \end{pmatrix} \tag{9}
$$

It should be mentioned that *a* is set to be along [100], but *b* is along [010] or [001] depending on physical situations and c is consequently along [001] or [010]. The Rex texture can often be obtained without resorting to the above process because the AMSD//MYMD condition is so dominant that the Rex texture can be obtained by the following priority order.

The 1st priority: When AMSD is cristallographically the same as MYMD, No texture changes after Rex [10].

The 2nd priority: When AMSD crystallographically differs from MYMD, the Rex texture is determined such that AMSD in the matrix is parallel to MYMD in the Rexed grain, with one common axis of rotation between the deformed and Rexed states. The common axis can be ND, TD, or other direction (e.g. <110> for bcc metals). This may be related to minimum atomic movement at the AMSD//MYMD constraints. However, we do not know the exact physical picture of this.

The 3rd priority: When the first two conditions are not met, the method explained to obtain Eq. 9 is used.

## **3. Electrodeposits and vapor-deposits**

rotates during deformation, the absolute value of shear strain rates |*dγ*(*i*) */dε*| on slip systems *i* can vary with strain *ε* of specimen. For a strain up to *ε* = *e*, the contribution of each slip system

> ee

The above equation is illustrated in Figure 5. If a deformation texture is stable, the shear strain

So far methods of obtaining AMSD have been discussed. This is good enough for prediction of fiber textures. However, the stress states around dislocation arrays are not uniaxial but triaxial. Unfortunately we do not know the stress fields of individual dislocations in real crystals, but know Burgers vectors. Therefore, AMSD obtained above applies to real crystals. Any stress state has three principal stresses and hence three principal stress directions which are perpendicular to each other. Once we know the three principal stress directions, the Rex textures are determined such that the three directions in the deformed matrix are parallel to three <100> directions in the Rexed grain, when MYMDs are <100>. In figure 6, let the unit

the unit vector *a* referred to the crystal coordinate system. AMSD is one of three principal stress

0 *e*

**A** AMSD**)(** 

If the unit vectors *a*, *b*, and *c* are set to be along [100], [010], and [001] after Rex, components of the unit vectors are direction cosines relating the deformed and Rexed crystal coordinate systems, when MYMDs are <100>. That is, the (*hkl*)[*uvw*] deformation orientation is calculated

to transform to the (*hr kr lr*)[*ur vr wr*] Rex orientation using the following equation.

e

*ddd* ee *<sup>e</sup> ii* ò 0 ( () ) /

**AS** ) with 90 nearest to directions slip of one(

*d dd* ò (8)

are direction cosines of

() () <sup>0</sup> / *<sup>e</sup> i i*

 

vectors of **A**, **B**, and **C** be *a* [*a*<sup>1</sup> *a*<sup>2</sup> *a*3], *b* [*b*<sup>1</sup> *b*<sup>2</sup> *b*3], and *c* [*c*<sup>1</sup> *c*<sup>2</sup> *c*3], where *ai*

e

*d d <sup>i</sup>*)(

**BAC**

**SAC**

**Figure 6.** Relationship between three principal stress directions **A**, **B**, and **C**.

directions. Two other principal stresses are obtained as explained in Figure 6.

for crystal rotation during deformation up to ε = *e.*

rates on the slip systems are independent of deformation.

to AMSD is proportional to

8 Recent Developments in the Study of Recrystallization

**Figure 5.** Calculation of γ(*i*)

When the density of dislocations in electrodeposits and vapor deposits is high, the deposits undergo Rex when annealed. AMSD in the deposits can be determined by their textures. The density of dislocations whose Burgers vectors are directed away from the growth direction (GD)of deposits was supposed to be higher than when the Burgers vector is nearly parallel to GD because dislocations whose Burgers vector is close to GD are easy to glide out from the deposits by the image force during their growth [11]. This was experimentally proved in a Cu electrodeposit with the <111> orientation [12]. Therefore, AMSDs are along the Burgers vectors nearly normal to GD.

#### **3.1. Copper, nickel, and silver electrodeposits**

Lee et al. found that the <100>, <111>, and <110> textures (inverse pole figures: IPFs) of Cu electrodeposits which were obtained from Cu sulfate and Cu fluoborate baths [13,14], and a cyanide bath [15] changed to the <100>, <100>, and <√310> textures, respectively, after Rex as shown in Figure 7. The texture fraction (TF) of the (*hkl*) reflection plane is defined as follows:

$$\text{TF}(hkl) = \frac{\text{I}(hkl) / \text{I}\_{\text{o}}(hkl)}{\sum \left[ \text{I}(hkl) / \text{I}\_{\text{o}}(hkl) \right]} \tag{10}$$

where I(*hkl*) and Io(*hkl*) are the integrated intensities of (*hkl*) reflections measured by x-ray diffraction for an experimental specimen and a standard powder sample, respectively, and Σ means the summation. When TF of any (*hkl*) plane is larger than the mean value of TFs, a preferred orientation or a texture exists in which grains are oriented with their (*hkl*) planes parallel to the surface, or with their <*hkl*> directions normal to the surface. When TFs of all reflections are the same, the distribution of crystal orientation is random. TFs of all the reflections sum up to unity. Figure 7 indicates that the deposition texture of <100> remains unchanged after Rex. This is expressed as <100>*D*→<100>*R*. All the samples were freestanding and so subjected to no external external stresses during annnealing. The results are explained by SERM in Section 2. We have to know MYMD of Cu and AMSDs of Cu electrodeposits. Young's modulus *E* of cubic crystals can be calculated using Eq. 11 [16].

For fcc Cu, S11=0.018908, S44 =0.016051, S12 = -0.008119 GPa-1 at 800 K [17], which in turn gives rise to [*S*44-2(*S*11-*S*12)] < 0, and so MYMDs are <100>. MYMDs and the Burgers vectors of Cu are along the <100> directions and the <110> directions, respectively. There are six equivalent directions in the <110> directions, with opposite directions being taken as the same. As already

For the <100> oriented Cu (simply <100> Cu) deposit, two of the six <110> directions are at 90°

which is AMSD, change to the <100> directions after Rex, resulting in the <100> Rex texture

For the <111> Cu deposit, three of the six <110> directions are at right angles with the [111] GD;

three <110> directions, AMSD, can change to <100> after Rex, but angles between the <110> di‐

<110> directions in as-deposited grains and the <100> directions in Rexed grains is therefore im‐ possible in a grain. Two of the <110> directions in neighboring grains, which are at right angles with each other, can change to the <100> directions to form the <100> nuclei in grain bounda‐ ries, which grow at the expense of high energy region, as shown in Figure 9b. Thus, the <111> deposition texture change to the <100> Rex texture, in agreement with the measured result.

> <110> <110>

**Figure 8.** Drawings explaining that <100> deposition texture (a) remains unchanged after Rex (b).

a b

ings for explanation of <111> deposition to <100> Rex texture transformation.

<100>

a growth direction b

110

and the angle between the <100> directions is 90°. Correspondence between the

<100>

<100>

grain boundary

<sup>111</sup> <sup>111</sup>

110

**Figure 9.** (a) <110> directions in <111> oriented fcc crystal in which arrow indicates [111] growth direction. (b) Draw‐

100

<100>

with GD, as shown in Figure 8. The two <110> directions,

with GD, as shown in Figure 9 a. The former

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11

grain Rexed

111 111

100

100

explained, AMSD is along the Burgers vector which is approximately normal to GD.

and the remaining four are at 45o

rections are 60o

(Figure 8b) in agreement with the experimental result.

the remaining three <110> directions are at 35.26o

**Figure 7.** Deposition and Rex textures of Cu electrodeposits. <*hkl*>*D*→<*uvw*>*<sup>R</sup>* means that <*hkl*> deposition texture changes to <*uvw*> Rex texture. For <100>*D*, Rex peaks are shifted rightward by 1º from their original positions to be distinguished from deposition peaks. TF data [13] and IPFs [14].

$$1/E = \mathcal{S}\_{11} + [\mathcal{S}\_{44} - \mathcal{Q}(\mathcal{S}\_{11} - \mathcal{S}\_{12})](a\_{11}^2 a\_{12}^2 + a\_{12}^2 a\_{13}^2 + a\_{13}^2 a\_{11}^2) \tag{11}$$

where *S*ij are compliances and *a*1i are the direction cosines relating the uniaxial stress direction *x* ′ 1 to the symmetry axes *xi* . When [*S*44-2(*S*11-*S*12)] < 0, or *A*=2(*S*11-*S*12)/*S*<sup>44</sup> > 1, (*a*<sup>11</sup> <sup>2</sup> *<sup>a</sup>*<sup>12</sup> <sup>2</sup> <sup>+</sup> *<sup>a</sup>*<sup>12</sup> <sup>2</sup> *<sup>a</sup>*<sup>13</sup> <sup>2</sup> <sup>+</sup> *<sup>a</sup>*<sup>13</sup> <sup>2</sup> *<sup>a</sup>*<sup>11</sup> 2 ) = 0 yields the minimum Young's modulus, which is obtained at *a*11 = *a*12 = *a*13 = 0. Therefore, MYMDs are parallel to <100>. When [*S*44-2(*S*11-*S*12)]>0, or *A*< 1, the maximum value of (*a*<sup>11</sup> <sup>2</sup> *<sup>a</sup>*<sup>12</sup> <sup>2</sup> <sup>+</sup> *<sup>a</sup>*<sup>12</sup> <sup>2</sup> *<sup>a</sup>*<sup>13</sup> <sup>2</sup> <sup>+</sup> *<sup>a</sup>*<sup>13</sup> <sup>2</sup> *<sup>a</sup>*<sup>11</sup> <sup>2</sup> ) yields the minimum Young's modulus, which is obtained at *a*11 <sup>2</sup> <sup>=</sup>*a*<sup>12</sup> <sup>2</sup> <sup>=</sup>*a*<sup>13</sup> <sup>2</sup> =1 / 3. Therefore, MYMDs are parallel to <111>. When [*S*44-2(*S*11-*S*12)] = 0, or *A* = 1, *E* is independent of direction, in other words, the elastic properties are isotropic. *A* is usually referred to as Zener's anisotropy factor. Summarizing, MYMDs // <100> for *A*>1, MYMDs// <111> for *A*<1, and elastic isotropy for *A*=1.

For fcc Cu, S11=0.018908, S44 =0.016051, S12 = -0.008119 GPa-1 at 800 K [17], which in turn gives rise to [*S*44-2(*S*11-*S*12)] < 0, and so MYMDs are <100>. MYMDs and the Burgers vectors of Cu are along the <100> directions and the <110> directions, respectively. There are six equivalent directions in the <110> directions, with opposite directions being taken as the same. As already explained, AMSD is along the Burgers vector which is approximately normal to GD.

where I(*hkl*) and Io(*hkl*) are the integrated intensities of (*hkl*) reflections measured by x-ray diffraction for an experimental specimen and a standard powder sample, respectively, and Σ means the summation. When TF of any (*hkl*) plane is larger than the mean value of TFs, a preferred orientation or a texture exists in which grains are oriented with their (*hkl*) planes parallel to the surface, or with their <*hkl*> directions normal to the surface. When TFs of all reflections are the same, the distribution of crystal orientation is random. TFs of all the reflections sum up to unity. Figure 7 indicates that the deposition texture of <100> remains unchanged after Rex. This is expressed as <100>*D*→<100>*R*. All the samples were freestanding and so subjected to no external external stresses during annnealing. The results are explained by SERM in Section 2. We have to know MYMD of Cu and AMSDs of Cu electrodeposits.

*RD* <sup>100</sup> <sup>100</sup> *RD* <sup>111</sup> <sup>100</sup> *RD* <sup>110</sup> <sup>103</sup>

22 22 22 11 44 11 12 11 12 12 13 13 11 1 / [ 2( )]( *E S S S S aa aa aa* - - ) (11)

<sup>2</sup> ) yields the minimum Young's modulus, which is obtained at

<sup>2</sup> *<sup>a</sup>*<sup>12</sup> <sup>2</sup> <sup>+</sup> *<sup>a</sup>*<sup>12</sup> <sup>2</sup> *<sup>a</sup>*<sup>13</sup> <sup>2</sup> <sup>+</sup> *<sup>a</sup>*<sup>13</sup> <sup>2</sup> *<sup>a</sup>*<sup>11</sup> 2 )

**Figure 7.** Deposition and Rex textures of Cu electrodeposits. <*hkl*>*D*→<*uvw*>*<sup>R</sup>* means that <*hkl*> deposition texture changes to <*uvw*> Rex texture. For <100>*D*, Rex peaks are shifted rightward by 1º from their original positions to be

where *S*ij are compliances and *a*1i are the direction cosines relating the uniaxial stress direction

= 0 yields the minimum Young's modulus, which is obtained at *a*11 = *a*12 = *a*13 = 0. Therefore, MYMDs are parallel to <100>. When [*S*44-2(*S*11-*S*12)]>0, or *A*< 1, the maximum value of

is independent of direction, in other words, the elastic properties are isotropic. *A* is usually referred to as Zener's anisotropy factor. Summarizing, MYMDs // <100> for *A*>1, MYMDs//

. When [*S*44-2(*S*11-*S*12)] < 0, or *A*=2(*S*11-*S*12)/*S*<sup>44</sup> > 1, (*a*<sup>11</sup>

<sup>2</sup> =1 / 3. Therefore, MYMDs are parallel to <111>. When [*S*44-2(*S*11-*S*12)] = 0, or *A* = 1, *E*

Young's modulus *E* of cubic crystals can be calculated using Eq. 11 [16].

distinguished from deposition peaks. TF data [13] and IPFs [14].

10 Recent Developments in the Study of Recrystallization

*x* ′

(*a*<sup>11</sup> <sup>2</sup> *<sup>a</sup>*<sup>12</sup> <sup>2</sup> <sup>+</sup> *<sup>a</sup>*<sup>12</sup> <sup>2</sup> *<sup>a</sup>*<sup>13</sup> <sup>2</sup> <sup>+</sup> *<sup>a</sup>*<sup>13</sup> <sup>2</sup> *<sup>a</sup>*<sup>11</sup>

*a*11 <sup>2</sup> <sup>=</sup>*a*<sup>12</sup> <sup>2</sup> <sup>=</sup>*a*<sup>13</sup>

1 to the symmetry axes *xi*

<111> for *A*<1, and elastic isotropy for *A*=1.

For the <100> oriented Cu (simply <100> Cu) deposit, two of the six <110> directions are at 90° and the remaining four are at 45o with GD, as shown in Figure 8. The two <110> directions, which is AMSD, change to the <100> directions after Rex, resulting in the <100> Rex texture (Figure 8b) in agreement with the experimental result.

For the <111> Cu deposit, three of the six <110> directions are at right angles with the [111] GD; the remaining three <110> directions are at 35.26o with GD, as shown in Figure 9 a. The former three <110> directions, AMSD, can change to <100> after Rex, but angles between the <110> di‐ rections are 60o and the angle between the <100> directions is 90°. Correspondence between the <110> directions in as-deposited grains and the <100> directions in Rexed grains is therefore im‐ possible in a grain. Two of the <110> directions in neighboring grains, which are at right angles with each other, can change to the <100> directions to form the <100> nuclei in grain bounda‐ ries, which grow at the expense of high energy region, as shown in Figure 9b. Thus, the <111> deposition texture change to the <100> Rex texture, in agreement with the measured result.

**Figure 8.** Drawings explaining that <100> deposition texture (a) remains unchanged after Rex (b).

**Figure 9.** (a) <110> directions in <111> oriented fcc crystal in which arrow indicates [111] growth direction. (b) Draw‐ ings for explanation of <111> deposition to <100> Rex texture transformation.

posits. Figure 11 shows four different deposition and corresponding Rex textures of Ag electrodeposits. Samples a, b, and c shows results similar to Cu electrodeposits, except that minor <221> component, which is the primary twin component of the <100> component in the Rex textures, is stronger than that of Cu deposits. The strong development of twins in Ag is

<sup>111</sup> levels:

a b c d

The deposition texture of Sample d was well described by 0.32<112> + 0.14<127>T + 0.25<113> + 0.23<557>T + 0.06<19 19 13>TT with each of individual orientations being superimposed with a Gaussian peak of 8°. Here <127>T indicates the twin orientation of its preceeding <112> orientation, and TT indicates secondary twin. Thus, the main components in deposition texture of Sample d are <112>, <113>, and <557>. The <110> directions that are nearly normal to GD will be AMSD and in turn determine the Rex texture. Table 1 gives angles between <110> and [11w]. Table 1 shows that the probability of <110> directions being normal to GD is the highest. The <110> directions normal to GD will become parallel to the <100> directions (MYMS) after Rex. Therefore, the Rex texture will be the <100> orientation for the same reason as in the <111>

Table 2 shows TFs (Eq. 10) of Cr electrodeposits obtained under three electrodeposition conditions. Specimen Cr-A has a strong <111> fiber texture. The texture of Cr-B is characterized by weak <111>, and that of Cr-C is by weak <100>. The optical microstructure and hardness test results and others indicated that all the specimens were fully Rexed at 1173 K. TFs as functions of annealing temperature and time in Figure 12 indicate that the deposition texture of Cr-A little change after Rex. The pole figures in Figures 13 and 14 indicate the deposition textures of Cr-B and Cr-C little change after Rex. In conclusion, the <100> and <111> deposition textures of Cr electrodeposits little change after Rex. These results are compatible with SERM as discussed in what follows. There are four equivalent <111> directions in bcc Cr crystal, with opposite directions being taken as the same. For the <111> Cr deposit, one of four <111>

100

1,1.3,1.6,1.9,2.2 max.2.5

levels: 1,2,3 max. 3.6

110

110

111

100 110 100 110

2,3.2,4.4,5.6,6, 8.8,9.2, 10.4 max.10.8

levels: 1,1.5,2,2.5,3 max. 3.2

100 110

111

111

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13

110

111

due to its lower stacking fault energy (~22 mJm-2) than that of Cu (~80 mJm-2).

2,3.5,5,6.5,8, 8.5,10,11.5,13,1 4.5 max 14.7

**Figure 11.** Deposition (top) and Rex (bottom) textures (IPFs) of Ag electrodeposits [22].

100 110

2,4,6,8,10 max. 11.6

<sup>111</sup> density levels:

111

111

1,1.5,2,2.5,3 max 3.4

1.5,2,2.5,3, 3.5,4,4.5 max.4.5

orientation of the deposit [22].

**3.2. Chromium electrodeposits**

100

100 110

100

110

**Figure 10.** directions in [110] oriented fcc crystal.

For the <110> Cu deposit, one <110> direction is normal to the <110> GD and the remaining four <110> directions are at 60o with the <110> GD, as shown in Figure 10. The first one of the <110> directions and the last four <110> directions are likely to determine the Rex texture because the last four directions are closer to the deposit surface than to GD. Recalling that the <110> directions change to <100> directions after Rex, GD of Rexed grains should be at 60o and 90o with the <100> directions, MYMD, at the same time. GD satisfying the condition is <√310>, in agreement with the experimental results.

So far we have discussed the evolution of the Rex textures from simple deposition textures. A Cu deposit whose texture can be be approximated by a weak duplex texture consisting of the <111> and <110> orientations developed the Rex texture which is approximated by a weak <√310> orientation rather than <100> + <√310> [18]. For the duplex deposition texture, the Rex texture may not consist of the Rex orientation components from the deposition orientation components because differently oriented grains can have different energies. The tensile strengths of copper electrodeposits showed that the tensile strength of the specimens with the <110> texture was higher than those with the <111> texture obtained from the similar electro‐ deposition condition. This implies that the <110> specimen has the higher defect densities than the <111> specimen [18,19]. Therefore, the <110> grains are likely to have higher driving force for Rex than the <111> grains, resulting in the <√310> texture after Rex, in agreement with experimental result [18].

For Ni, S11= 0.009327, S<sup>44</sup> = 0.009452, S12 = -0.003694 GPa-1 at 760 K [20], which in turn gives rise to [*S*44-2(*S*11-*S*12)] < 0, and so MYMDs are <100>. Therefore, the deposition to Rex texture transformation of Ni electrodeposits is expected to be similar to that of Cu electrodeposits. As expected, freestanding Ni electrodeposits of 30-50 µm in thickness showed that the <100> deposition texture remained unchanged after Rex, and the <110> deposition texture changed to <√310> after Rex [21].

For Ag, S11= 0.03018, S44 = 0.02639, S12 = -0.0133 GPa-1 at 750 K [17], which in turn gives [*S*44-2(*S*11-*S*12)] < 0, and so MYMDs are <100>. Therefore, the deposition to Rex texture trans‐ formation of freestanding Ag electrodeposits is expected to be similar to that of Cu electrode‐ posits. Figure 11 shows four different deposition and corresponding Rex textures of Ag electrodeposits. Samples a, b, and c shows results similar to Cu electrodeposits, except that minor <221> component, which is the primary twin component of the <100> component in the Rex textures, is stronger than that of Cu deposits. The strong development of twins in Ag is due to its lower stacking fault energy (~22 mJm-2) than that of Cu (~80 mJm-2).

**Figure 11.** Deposition (top) and Rex (bottom) textures (IPFs) of Ag electrodeposits [22].

The deposition texture of Sample d was well described by 0.32<112> + 0.14<127>T + 0.25<113> + 0.23<557>T + 0.06<19 19 13>TT with each of individual orientations being superimposed with a Gaussian peak of 8°. Here <127>T indicates the twin orientation of its preceeding <112> orientation, and TT indicates secondary twin. Thus, the main components in deposition texture of Sample d are <112>, <113>, and <557>. The <110> directions that are nearly normal to GD will be AMSD and in turn determine the Rex texture. Table 1 gives angles between <110> and [11w]. Table 1 shows that the probability of <110> directions being normal to GD is the highest. The <110> directions normal to GD will become parallel to the <100> directions (MYMS) after Rex. Therefore, the Rex texture will be the <100> orientation for the same reason as in the <111> orientation of the deposit [22].

#### **3.2. Chromium electrodeposits**

**Figure 10.** directions in [110] oriented fcc crystal.

12 Recent Developments in the Study of Recrystallization

in agreement with the experimental results.

four <110> directions are at 60o

experimental result [18].

to <√310> after Rex [21].

For the <110> Cu deposit, one <110> direction is normal to the <110> GD and the remaining

<110> directions and the last four <110> directions are likely to determine the Rex texture because the last four directions are closer to the deposit surface than to GD. Recalling that the <110> directions change to <100> directions after Rex, GD of Rexed grains should be at 60o

90o with the <100> directions, MYMD, at the same time. GD satisfying the condition is <√310>,

So far we have discussed the evolution of the Rex textures from simple deposition textures. A Cu deposit whose texture can be be approximated by a weak duplex texture consisting of the <111> and <110> orientations developed the Rex texture which is approximated by a weak <√310> orientation rather than <100> + <√310> [18]. For the duplex deposition texture, the Rex texture may not consist of the Rex orientation components from the deposition orientation components because differently oriented grains can have different energies. The tensile strengths of copper electrodeposits showed that the tensile strength of the specimens with the <110> texture was higher than those with the <111> texture obtained from the similar electro‐ deposition condition. This implies that the <110> specimen has the higher defect densities than the <111> specimen [18,19]. Therefore, the <110> grains are likely to have higher driving force for Rex than the <111> grains, resulting in the <√310> texture after Rex, in agreement with

For Ni, S11= 0.009327, S<sup>44</sup> = 0.009452, S12 = -0.003694 GPa-1 at 760 K [20], which in turn gives rise to [*S*44-2(*S*11-*S*12)] < 0, and so MYMDs are <100>. Therefore, the deposition to Rex texture transformation of Ni electrodeposits is expected to be similar to that of Cu electrodeposits. As expected, freestanding Ni electrodeposits of 30-50 µm in thickness showed that the <100> deposition texture remained unchanged after Rex, and the <110> deposition texture changed

For Ag, S11= 0.03018, S44 = 0.02639, S12 = -0.0133 GPa-1 at 750 K [17], which in turn gives [*S*44-2(*S*11-*S*12)] < 0, and so MYMDs are <100>. Therefore, the deposition to Rex texture trans‐ formation of freestanding Ag electrodeposits is expected to be similar to that of Cu electrode‐

with the <110> GD, as shown in Figure 10. The first one of the

and

Table 2 shows TFs (Eq. 10) of Cr electrodeposits obtained under three electrodeposition conditions. Specimen Cr-A has a strong <111> fiber texture. The texture of Cr-B is characterized by weak <111>, and that of Cr-C is by weak <100>. The optical microstructure and hardness test results and others indicated that all the specimens were fully Rexed at 1173 K. TFs as functions of annealing temperature and time in Figure 12 indicate that the deposition texture of Cr-A little change after Rex. The pole figures in Figures 13 and 14 indicate the deposition textures of Cr-B and Cr-C little change after Rex. In conclusion, the <100> and <111> deposition textures of Cr electrodeposits little change after Rex. These results are compatible with SERM as discussed in what follows. There are four equivalent <111> directions in bcc Cr crystal, with opposite directions being taken as the same. For the <111> Cr deposit, one of four <111> directions is along GD and the remaining three <111> directions are at an angle of 70.5o with GD (Figure 15). The remaining three <111> directions can be AMSDs. They will become parallel to MYMDs of Rexed grains. The compliances of Cr are *S*11 =0.00314, *S*44 = 0.0101, *S*12 = -0.000567 GP-1 at 500 K [23], which lead to [*S*44-2(*S*11-*S*12)] > 0. Therefore, MYMDs of Cr are <111>, which are also AMSDs of the deposit. Therefore, the <111> and <100> textures of Cr deposits do not change after Rex, as can be seen from Figure 15, in agreement with experimental results.


**Table 1.** Angles between <110> and [11w] directions (°)


**Table 2.** Texture fractions (TF) of reflection planes of Cr electrodeposits A, B, and C [14]. Bold-faced numbers indicate highest TFs in corresponding deposits.

gas purity, and substrate bias. The 3.81 cm diameter target was made from commercial grade OFHC forged Cu-bar stock containing approximately 100 ppm oxygen by weight with only traces of other elements. The substrates were 2.54 cm diameter by 6.2 mm thick disks made of OFHC Cu. These disks were electron beam welded to a stainless-steel tube to provide direct water-cooling for temperature control during sputtering. As-deposited grains were approxi‐ mately 100 nm in diameter. Room-temperature Rex and grain growth displaying no twins were observed approximately 9 h after removal from the sputtering apparatus. Nucleation sites were almost randomly distributed. Hardness of the unrecrystallized matrix remained at ~230 DPH from the time it was sputtered until Rex, when it abruptly dropped to approximately 60 DPH in the Rexed grains. Rex resulted in a texture transformation from the <111> deposition texture to the <100> Rex texture. Since the substrate is also Cu, the orientation transition from <111> to <100> cannot be attributed to thermal strains. The driving force for Rex must be the

**Figure 13.** (200) pole figures of Cr-B (left) before and (right) after annealing at 1173 K for 1 h [14].

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15

**Figure 14.** (200) pole figures of Cr-C (left) before and (right) after annealing at 1173 K for 1 h [14].

**Figure 12.** TFs of Cr-A as functions of annealing (a) temperature for 1 h and (b) time at 903 K [14].

#### **3.3. Copper and silver vapor-deposits**

Patten et al. [24] formed deposits of Cu up to 1mm in thickness at room temperature in a triode sputtering apparatus using a krypton discharge under various conditions of sputtering rate,

**Figure 13.** (200) pole figures of Cr-B (left) before and (right) after annealing at 1173 K for 1 h [14].

directions is along GD and the remaining three <111> directions are at an angle of 70.5o

GD (Figure 15). The remaining three <111> directions can be AMSDs. They will become parallel to MYMDs of Rexed grains. The compliances of Cr are *S*11 =0.00314, *S*44 = 0.0101, *S*12 = -0.000567 GP-1 at 500 K [23], which lead to [*S*44-2(*S*11-*S*12)] > 0. Therefore, MYMDs of Cr are <111>, which are also AMSDs of the deposit. Therefore, the <111> and <100> textures of Cr deposits do not change after Rex, as can be seen from Figure 15, in agreement with experimental results.

**110 -110 101 -101 011 0-1 1**

557 44.7 90 31.5 81.8 31.5 81.8 112 54.7 90 30 73.2 30 73.2 113 64.8 90 31.5 64.8 31.5 64.8

**(110) (200) (211) (220) (310) (222) Texture**

**Table 2.** Texture fractions (TF) of reflection planes of Cr electrodeposits A, B, and C [14]. Bold-faced numbers indicate

0.0

0.2

0.4

**Texture fraction**

Patten et al. [24] formed deposits of Cu up to 1mm in thickness at room temperature in a triode sputtering apparatus using a krypton discharge under various conditions of sputtering rate,

0.6

0.8

1.0

0 100 200 300 400 500

**Annealing time (min.)**

 (110) (200) (211) (222)

Cr-A 0.02 0.05 0 0 0 **0.93** Strong <111>

 (110) (200) (211) (222)

Cr-B 0.03 0.15 0.28 0 0.01 **0.53** <111> Cr-C 0.19 **0.47** 0.13 0.05 0.13 0.03 <100>

200 400 600 800 1000 1200 1400

**Annealing temp. (K)**

**Figure 12.** TFs of Cr-A as functions of annealing (a) temperature for 1 h and (b) time at 903 K [14].

**Table 1.** Angles between <110> and [11w] directions (°)

14 Recent Developments in the Study of Recrystallization

highest TFs in corresponding deposits.

0.0

**3.3. Copper and silver vapor-deposits**

0.2

0.4

**Texture fraction**

0.6

0.8

1.0

with

**Figure 14.** (200) pole figures of Cr-C (left) before and (right) after annealing at 1173 K for 1 h [14].

gas purity, and substrate bias. The 3.81 cm diameter target was made from commercial grade OFHC forged Cu-bar stock containing approximately 100 ppm oxygen by weight with only traces of other elements. The substrates were 2.54 cm diameter by 6.2 mm thick disks made of OFHC Cu. These disks were electron beam welded to a stainless-steel tube to provide direct water-cooling for temperature control during sputtering. As-deposited grains were approxi‐ mately 100 nm in diameter. Room-temperature Rex and grain growth displaying no twins were observed approximately 9 h after removal from the sputtering apparatus. Nucleation sites were almost randomly distributed. Hardness of the unrecrystallized matrix remained at ~230 DPH from the time it was sputtered until Rex, when it abruptly dropped to approximately 60 DPH in the Rexed grains. Rex resulted in a texture transformation from the <111> deposition texture to the <100> Rex texture. Since the substrate is also Cu, the orientation transition from <111> to <100> cannot be attributed to thermal strains. The driving force for Rex must be the internal stress due to defects such as vacancies and dislocations. Therefore, the texture transition is consistent with the prediction of SERM.

**4. Axisymmetrically drawn fcc metals**

orientation during annealing.

111

110 contour 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 :level

cal-dies of 9° in half-die angle, taking strain-hardening per pass into count [31].

**Figure 16.** Calculated IPFs in centeral axis zone of Cu wire drawn by 90% in 14 passes (~15% per pass) through coni‐

passes 4 passes 8 passes 10 passes 12 passes 14

100

It is known that the texture of axisymmetrically drawn fcc metals is characterized by major <111> + minor <100> components, and the drawing texture changes to the <100> texture after Rex [29,30]. Figure 16 shows calculated textures in the center region of 90% drawn copper wire taking work hardening per pass into account. The drawing to Rex texture transition was explained by SERM [4]. Since the drawing texture is stable, we consider the [111] and [100] fcc crystals representing the <111> and <100> fiber orientations constituting the texture. Figure 17 shows tetrahedron and octahedron consisting of slip planes (triangles) and slip directions (edges) for the [111] and [100] fcc crystals. The slip planes are not indexed to avoid complica‐ tion. The slip-plane index can be calculated by the vector product of two of three slip directions (edges) of a triangle constituting the slip-plane triangle. It follows from Figure 17a that three active slip directions that are skew to the [111] axial direction are [101], [110], and [011]. It should be noted that these directions are chosen to be at acute angles with the [111] direction (Section 2). Therefore, AMSD // ([101] + [110] + [011]) = [222] // [111]. That is, AMSD is along the axial direction. According to SERM, AMSD in the deformed matrix is along MYMD in the Rexed grain. MYMDs of most of fcc metals are <100>. Therefore, the <111> drawing texture changes to the <100> Rex texture. Now, the evolution of <100> Rex texture in the <100> deformed matrix is explained. Eight active slip systems in fcc crystal elongated along the [100] direction are calculated to be (111)[1 0-1], (-111)[101], (1-1 1)[110], (1 1-1)[1-1 0], (111)[1-1 0], (-111)[110], (1-1 1)[10-1], and (1 1-1)[101], if the slip systems are {111}<110> [32]. It is noted that the slip directions are chosen to be at acute angles with the [100] axial direction. These slip systems are shown in Figure 17 b. AMSD is obtained, from the vector sum of the active slip directions, to be parallel to [100], which is also MYMD of fcc metals. Therefore, the <100> drawing texture remains unchanged after Rex (1st priority in Section 2), and the <111> + <100> orientation changes to <100> after Rex, regardless of relative intensity of <111> to <100> in the deformation texture. The <100> grains in deformed fcc wires are likely to act as nuclei for Rex. The texture change during annealing might take place by the following process. The <100> grains retain their deformation texture during annealing by continuous Rex, or by recoverycontrolled processes, without long-range high-angle boundary migration. The <100> grains grow at the expense of their neighboring <111> grains that are destined to assume the <100>

Recrystallization Textures of Metals and Alloys

http://dx.doi.org/10.5772/54123

17

**Figure 15.** Thin arrows (AMSDs) and thick arrows (GD) in [111] and [001] Cr crystals.

Greiser et al. [25] measured the microstructure and texture of Ag thin films deposited on different substrates using DC magnetron sputtering under high vacuum conditions (base pressure: 10-8 mbar, partial Ar pressure during deposition: 10-3 mbar). A weak <111> texture in a 0.6 µm thick Ag film deposited on a (001) Si wafer with a 50 nm thermal SiO2 layer at room temperature becomes stronger with increasing thickness. It is generally accepted that a random polycrystalline structure is obtained up to a critical film thickness unless an epitaxial growth condition is satisfied. Therefore, the <111> texture developed in the 0.6 µm film was weak and became stronger with increasing thickness. This is consistent with the preferred growth model [26]. They also found that the texture of the film deposited at room temperature was "high <111>", whereas the texture of the film deposited at 200 °C was characterized by a low amount of the <111> component and a high amount of the random component. This is also consistent with the preferred growth model.

Post-deposition annealing was carried out in a vacuum furnace at 400 °C with a base pressure of 10-6 mbar, a partial H2 pressure of 10 mbar, and under environmental conditions. The postdeposition grain growth was the same for annealing in high vacuum and in environmental conditions. A dramatic difference in the extent of growth was recognized in the micrographs of the 0.6 and 2.4 µm thick films. The 0.6 µm thick film showed normally grown grains with the <111> orientation; the average grain size was about 1 to 2 µm. This can be understood in light of the surface energy minimization. In contrast, in 2.4 µm thick films, abnormally large grains with the <001> orientation were found. These grains grew into the matrix of <111> grains. The grain boundaries between the abnormally grown grains have a meander-like shape unlike the usual polygonal shape. They could not explain the results by the model of Carel, Thomson, and Frost [27]. According to the model, the strain energy minimization favors the growth of <100> grains. The growth mode should be affected by strain and should not be sensitive to the initial texture. These predictions are at variance with the experimental results in which freestanding, stress-free films also showed abnormal growth of giant grains with <001> texture. The 2.4 µm thick films deposited at 100 °C or below could have dislocations whose density was high enough to cause Rex, which in turn gave rise to the texture change from <111> to <001> regardless of the existence of substrate when annealed, as explained in the previous section. Thus, the <111> to <100> texture change in the 2.4 µm thick films is compatible with SERM [28].
