Keywords

1 Introduction

The static recrystallization (SRX) that occurs during annealing cold or hot worked metals causes dramatical change of microstructure. Fine or coarse, equiaxed or lath-like recrystallized grains together with various textures can be generated, which have a significant impact on the material’s performance [1]. The morphology and texture of recrystallized grains are determined by various factors such as deformed microstructure, second phase particles, and annealing conditions. For Al–Mn alloys, for example, studies [2, 3] have shown that high temperature annealing after cold rolling produces fine equiaxed recrystallized grains with Cube as the main texture component. However, during low temperature or non-isothermal annealing, dynamic precipitation brings a strong pinning effect on the boundary migration of nuclei, resulting in coarse lath-like recrystallized grains, with high strength of P and ND-rotated Cube textures, which is closely related to the high mobility of the special grain boundary (Σ7, 40° <111>) and the uneven distribution of the dispersed particles [4]. Many numerical approaches such as cellular automaton [5, 6] (CA), Monte Carlo [7], and phase field [8, 9] models have been developed to simulate the microstructural evolution over the thermo-mechanical processes like SRX. However, there is no model so far to consider the anisotropy of recrystallized grain growth, which may come from the fibrous morphology of deformed grains, the orientation dependence of grain boundary mobility and the inhomogeneous distribution of dispersed particles.

The CA model has gained popularity because of its length scale calibrations and high efficiency. The core algorithms of the CA technique mainly include discrete spatial and temporal evolution of complex systems by applying local or global probabilistic or deterministic transformation rules on the location of a lattice. In this work, oriented growth and anisotropic dispersoid pinning effects on boundary migration are incorporated into a CA model to simulate the morphology evolution of recrystallized grain in cold-rolled aluminum alloy. Diverse grain shape and textures can be predicted when varying the parameters of annealing condition, and the underlying physical mechanism is discussed based on the simulated results.

2 Model Setup

2.1 Cellular Automaton Framework

A cellular automaton with periodic borders and a cubic grid is employed in this work. State variables such as Euler angles, grain number, and dislocation density are assigned to each cell. The crystallographic misorientation between each cell and its neighbors is calculated in every step, and the grain boundary is regarded as existing when the misorientation angle is larger than a critical value. Whether the cell at grain boundary is transformed to the target neighbors is determined by the transition state variable:

$$ f_{{{\text{tran}}}} = \sum\limits_{i = 1}^{{N_{{{\text{nbrs}}}} }} {G_{i} \frac{{v_{i} \Delta t}}{{l_{0} }}} $$
(1)

where Nnbrs is the total number of all von Neumann neighbor cells (6 in 3D space), vi is the velocity at which the neighbor i encroaches the central cell of interest, i.e., migration rate, Δt is the time increment, l0 is the cell size, Gi is a logical factor that is assigned 1 only if the maximum vi of all neighbors is greater than 0 and the cell of neighbor i belongs to the same grain as that with the maximum vi. Otherwise, Gi is 0. The value of ftran of all cells will be saved and followed in each step. As ftran reaches 1, cell state variables like orientation, grain number, and dislocation density are switched, with the neighbor having the highest vi at present being the target. The ftran will be reset to 0 after the switch.

A time increment that adjusts dynamically with the maximal migration speed is used in this model:

$$ \Delta t = \frac{{l_{0} }}{{v_{\max } }} $$
(2)

where vmax is the highest migration velocity among all boundaries, which is not constant throughout the simulation process.

2.2 Boundary Migration Velocity

The migration rate v of grain boundary can be described by the following equation in accordance with Burke and Turnbull’s theory [10]:

$$ v = MP $$
(3)

where M and P are the boundary mobility and net driving pressure, respectively. M is a parameter related to boundary misorientation angle θ and temperature T, which can be expressed using the equation as below [11, 12]:

$$ M = M_{0} \exp \left( { - \frac{{Q_{b} }}{RT}} \right)\left( {1 - \exp \left( { - 5\left( {\frac{\theta }{{\theta_{m} }}} \right)^{4} } \right)} \right) $$
(4)

where M0 is the pre-factor, Qb is the activation energy for boundary diffusion, R is the general gas constant, θm is the defined critical misorientation angle for high angle boundary (15° is assumed here).

Equation (4) depicts the classical expression of grain boundary mobility, which only considers the relevance of misorientation angle, i.e., grain boundary mobility increases with misorientation angle from 0 to θm, and reaches saturation at θm. A new orientation dependent factor qs is introduced in this model, and a higher value of 5 is assigned for the Σ7 (40° <111>) boundary and 1 for the others. Hence, the pre-factor M0 in Eq. (4) is replaced by qsM0. Note that a boundary segment with rotation angle of 30~46° and rotation axis of ~15° to <111> (crystallographic direction) is regarded as Σ7 boundary in this work.

The driving pressure acting on the boundary can be calculated as follows:

$$ P = P_{s} + P_{c} + P_{z} $$
(5)

where Ps is the stored energy difference of grains on both sides of the boundary and can be formulated as

$$ P_{s} = 0.5\mu b^{2} \Delta \rho $$
(6)

where μ is the temperature dependent shear modulus, b is the length of Burgers vector, Δρ is the difference of dislocation density on both sides of grain boundary.

Pc refers to the driving force of curvature, which can be positive or negative, depending on the decrease or increase of local interface energy at grain boundaries. The Kremeyer model [13], given by Eqs. (7) and (8), is employed in this study because of its simplicity:

$$ P_{c} = \gamma_{s} \kappa $$
(7)
$$ \kappa = \frac{A}{{l_{0} }}\frac{{kink - N_{i} }}{N + 1} $$
(8)

where κ is the curvature obtained by Kremeyer approach, A is the shape factor (0.394 in 3D space), kink is the number of cells required to create a flat grain boundary (kink = 75 in 3D space), N is the total number of CA cells in a region containing two layers adjacent to the cell of interest (N = 124 in 3D space), and Ni is the number of cells belonging to the same grain with the central cell of interest in this region. γs is the boundary energy, and can be calculated using the Read-Shockley equation [14]:

$$ \gamma_{s} = \left\{ {\begin{array}{*{20}c} {\gamma_{m} ,\theta \ge \theta_{m} } \\ {\frac{{\gamma_{m} \theta }}{{\theta_{m} }}\left( {1 - \ln \frac{\theta }{{\theta_{m} }}} \right)} \\ \end{array} } \right.,\theta \le \theta_{m} $$
(9)

where γm is the energy of high angle boundary. A lower energy for 40° <111> boundary of aluminum by 5~10% compared with other “random” high angle boundaries has been reported [15]. As a result, a lower pinning effect by particles is expected since the pinning pressure is proportional to the boundary energy. Hence, in this study, a lower boundary energy \( \gamma ^{\prime}_{m} = 0.9\gamma _{m} \) is adopted on these special boundaries.

The last term Pz in Eq. (5) is the Zener drag pressure on boundary provided by the fine dispersive particles, and can be classically expressed as [16]:

$$ P_{z} = - \frac{{3f_{p} \gamma_{s} }}{{2r_{p} }} $$
(10)

where fp is the volume fraction of dispersive particles, rp is the average particle radius. Equation (10) gives a simplified but practical way to calculate the pinning effect of particles evenly distributed on the boundary. However, the particle distribution after plastic deformation, such as rolling, will be not uniform anymore and banded along the rolling direction. In addition, the axial ratio of particles is usually ignored since the orientation of the ellipsoidal axis prior to rolling is assumed to be random. However, following rolling deformation, the particles’ long axis is frequently getting parallel to the rolling direction. Both the anisotropy of the particle distribution and the orientation of the ellipsoid axis can result in a greater pinning resistance in the normal direction (ND) direction and less in the rolling direction (RD). Therefore, an anisotropic pinning factor wi is introduced into Eq. (10):

$$ P_{z - i} = - \frac{{3f_{p} \gamma_{s} }}{{2r_{p} w_{i} }} $$
(11)

where \(w_{i} = \exp (\beta e_{p} \varepsilon_{i} )\), β is a constant and here is 0.5, ep is the average axial ratio of particles, ε is the strain and i represents the three directions of RD, TD (transverse direction), and ND. Therefore, an anisotropic pinning force is imposed, and it becomes more pronounced with the increase of rolling strain and average axial ratio of particles.

2.3 Microstructure Initialization for Annealing

In order to simulate the recrystallized grain growth behavior during annealing, a reasonable cold-rolled microstructure must be initialized first, which mainly includes plate-like (or fibrous for 2D space) grain morphology and typical rolling textures such as Copper, S, and Brass. A virtual microstructure composed of fine equiaxed grains is created, and these grains are subsequently assigned rolling textures with banded or random distribution. Grains with orientations of Copper, S, and Brass (with a maximum random deviation of 5° from the standard orientation) account for around 40%, 30%, and 30% of the total respectively, as Fig. 1 shows (all the IPF maps in this paper are with respect to the ND direction). The physical dimensions of computational domains are set as 600 × 5 × 600 μm, with a cell size of 1 μm. The expansion of nuclei in the TD direction is disregarded, much fewer cell layers are thus set in this direction considering computational time cost.

Fig. 1
figure 1

The initialized cold-rolled microstructure. a Random orientations; b banded distribution of rolling textures; c random distribution of rolling textures

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The site saturation nucleation hypothesis is adopted in this paper, and a preset number and orientation of nuclei are artificially implanted at the beginning of annealing. The dislocation density in the deformed matrix is set to 5 × 1015 m−2, whereas that in the nuclei is 1 × 1010 m−2. The focus of this study is on the anisotropy of nuclei growth, regardless of the decrease of driving force caused by static recovery, although it will lead to the reduction of growth rate in the later stage of annealing. In this study, four examples with different annealing condition are calculated individually, as shown in Table 1. The setting for each annealing condition is described in detail in Sect. 3.

Table 1 Different settings of annealing condition
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3 Results and Discussion

In this investigation, the pre-factor M0 is 7.41 × 10–5 m·s−1·Pa−1, R is 8.314 J·mol−1·K−1, θm is 15°, μ is 20.8 GPa in 400 ℃, b is 2.86 × 10–10 m [17], γm is 0.324 J/m2 [16], and rolling strain ε is 3 (ε1 = 3, ε2 = 0, ε3 = −3). The volume fraction of dispersoid, if present, is set to 0.7%, and the average radius is 60 nm, with an average axial ratio of 1.2. It is assumed that nuclei occur at the beginning of annealing and the simulation stops as the volume fraction of recrystallization reaches 100%. Figures 2 and 3 present the final recrystallized microstructures of the four cases. It is evident that varying annealing conditions lead to different recrystallized grain sizes, shapes, and textures.

Fig. 2
figure 2

The simulated recrystallized microstructures in different annealing conditions. a–d IPF maps; e–h ODF sections (φ2 = 0°)

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Fig. 3
figure 3

Recrystallized grain morphology. a Axial ratio; b diameter

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3.1 Case 1 and 2: Oriented Growth Effect

Figure 2a reveals the recrystallized grains in Case 1, in which no dispersoid presents and 200 nuclei are given with orientations of 10% Cube, 10% Σ7 (with a misorientation of 40° <111> to the matrix), 10% rolling and 70% random. It can be seen from Figs. 2a and 3a that four different kinds of nuclei end up with distinct sizes, and the average size of nuclei with Σ7 orientation is the largest, following by Cube, random and rolling. The microstructures in Case 1 at recrystallization volume fractions of 9%, 25%, 75%, and 100% are depicted in Fig. 4. All types of nuclei can grow isotropically in the early stage of recrystallization when the grain size is small, although the growth rates are quite different. However, as the nuclei continue to grow, those having specific orientation relationship (Σ7) with neighboring matrix contact the original mother grain boundary in the ND direction, and the growth rate gets slower, while along the RD direction they can keep growing rapidly until encountering other nuclei, while there are a large number of smaller grains, still maintaining slow and isotropic growth. ODF sections in Fig. 4e–h show the textures changing from rolling to a common recrystallized pattern.

Fig. 4
figure 4

Simulated evolution of microstructure during annealing in Case 1. a-d IPF maps; e–h ODF sections (φ2 = 0°)

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A same annealing condition is set in Case 2, except that a random distribution of rolling textures is applied. Although the grain morphology in the fully recrystallized state (Figs. 2b and 3) is not significantly different from that in Case 1, the recrystallization growth process is quite distinct. The microstructures at recrystallization volume fractions of 25% in Cases 1–3 are shown in Fig. 5. As stated above, under the band-like distributed rolling textures, some nuclei having specific orientation relationship (Σ7) with neighboring matrix can grow quickly along the RD direction, whereas growth along the ND direction is constrained, resulting in recrystallized grain anisotropy. This is not observed in the random distributed rolling textures because the possibility of growing nuclei encountering specific grain boundaries in all directions is consistent. This further illustrates the effect of oriented growth on the shape and size of recrystallized grains.

Fig. 5
figure 5

The growing nuclei when the volume fraction of recrystallization is 25% in Cases 1–3 a–c IPF maps; d–f nuclei only

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3.2 Case 3 and 4: Anisotropic Pinning Effect

In order to study the effect of non-uniformly distributed dispersoid on the growth of recrystallized grains, particles with a constant volume fraction and size are considered in Case 3, on the basis of Case 1. It can be calculated by Eq. (10) that the pinning resistance in RD and ND direction is 0.008 MPa and 0.244 MPa, respectively. In comparison to Case 1, it produces a recrystallized structure compressed in the ND direction and elongated in the RD direction by applying the pinning resistance (Figs. 2c, 5c and f). The axial ratio of recrystallized grains increases (Fig. 3), indicating enhanced growth anisotropy.

In fact, if the homogenization treatment before rolling is insufficient for Al–Mn alloys, evident dynamic precipitation will occur during annealing, considerably inhibiting the quantity of nucleation. Only nuclei with those specific orientations (Σ7) can be kept and grow due to their higher mobility and less pinning by particles. Hence, in Case 4, only 30 nuclei are provided, and they are all with Σ7 orientations. Surprisingly but reasonably, abnormally coarse and lath-like grain structure is observed (Fig. 2d). Due to the extremely low nucleation rate, competition between nuclei is very low, and thus there is sufficient space for nuclei to expand. Under the simultaneous effect of oriented growth and anisotropic pinning resistance, abnormally coarse recrystallized grains with high axial ratio can be produced (Fig. 3).

In order to further explain the correlation between oriented growth and deformed structure, Fig. 6 displays the orientation relationship of coarse grains with the surrounding matrix during growth. It can be observed that these coarse grains frequently have a Σ7 relationship with the adjacent parent grains in the RD direction, while they are mostly “pinned” by the adjacent deformed matrix with other orientations in the ND direction. It is also possible to see the protrusions and retrusions of grain boundaries in the RD direction, which is quite compatible with the study of HUANG et al. [1].

Fig. 6
figure 6

The boundary map when the volume fraction of recrystallization is 25% in Case 4

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3.3 Recrystallized Textures

The most common textures after annealing in cold-rolled aluminum are Cube, ND-rotated Cube, P, rolling and random. If one rotates the deformed texture by 40° along all eight variants of <111> crystallographic directions, a transformed texture is obtained. Three different transformed textures are generated here by doing this operation for 500 orientations of Copper, S, and Brass, respectively, as shown in Fig. 7. It is indicated that the Copper deformation texture may encourage the development of P and ND-rotated Cube, while Cube and RD-rotated Cube after recrystallization can be enhanced by the presence of S and Brass, respectively. Figure 7d shows the transformed texture obtained by doing the rotation on randomly picked 500 orientations from the initialized rolling texture in Sect. 2.3. A potential recrystallized texture containing Cube, ND/RD-rotated Cube, and P can be observed. Therefore, it may be concluded that the recrystallized texture is determined by selective nucleation and oriented growth, given a certain deformed microstructure.

Fig. 7
figure 7

Textures transformed from: a copper; b S; c brass; d randomly-picked rolling orientations

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ODF sections at φ2 = 0 of all cases are shown in Fig. 2e–h. Strong Cube, ND-rotated Cube, and P textures are all observed, which are all attributed to the fast growing speed of those nuclei, and certainly, the fact that nuclei of those orientations are preset in this work. A more reasonable nucleation rate model that accounts for the mechanisms of thermal activation and selective nucleation is developed, which will be published in future work.

4 Conclusion

Based on the simulated results, it can be concluded that.

The oriented growth effect, which refers to the high mobility and lower particle pinning for special boundary, has been applied in the present CA model. Nuclei with a misorientation close to 40° <111> with the adjacent matrix can grow faster than those with other orientations. A lath-like grain shape can be obtained at the early stage of recrystallization because of the fibrous rolling microstructure and oriented growth effect, although the axial ratio may decrease afterwards due to the impingement of growing nuclei along RD direction.

The anisotropic pinning effect arises from the anisotropy of the particle distribution and orientation of the ellipsoid axis, resulting in a higher drag force along ND direction and lower along RD direction. A higher axial ratio of recrystallized grains can be obtained when both oriented growth and anisotropic pinning are present, especially for the case that nucleation is hindered.