Modeling Texture Evolution during Recrystallization in Aluminum

Modeling Texture Evolution during Recrystallization in Aluminum Abhijit Brahme1,2 , Joseph Fridy3 , Hasso Weiland3 and Anthony D Rollett1 1 Carnegie Mellon University, 5000 Forbes Avenue Pittsburgh, PA 15213-3890, 2 Currently at Max-Planck-Institute for Steel research, Max-Planck Str. 1, 40237 Dsseldorf, Germany, 3 Alcoa Technical Center, Alcoa Center, PA 15609 Abstract. The main aim of this work was to develop a model with predictive capability for micro structural evolution during recrystallization and to identify factors that exert the greatest effect on the development of texture. To achieve this aim, geometric and crystallographic observations from two orthogonal sections through a polycrystal were used as input to the computer simulations, to create a statistically representative three dimensional model. Assignment of orientations to the grains was performed so as to optimize agreement between the orientation (ODF) and misorientation (MDF) distributions of assigned and the observed orientations. The microstructures thus created were allowed to evolve using a Monte-Carlo simulation. As a demonstration of the model the effects of anisotropy, both in energy and in mobility, stored energy and oriented nucleation on overall texture development were studied. The results were analyzed with reference to the various established competing theories of oriented nucleation (ON) and oriented growth (OG). The results suggested that all of oriented nucleation, mobility anisotropy, stored energy and energy anisotropy (listed in order of their relative importance) influence texture development. It was also determined that comparison of simulated and measured textures throughout the recrystallization process is a more severe test of a model than the typical comparison of textures only at the end of the process.

Keywords: Modeling, Microstructure, Texture, Aluminum, Grain Boundary Mobility, Nucleation, Oriented Growth, Oriented Nucleation, Stored Energy, Monte Carlo Modeling.

Modeling Texture Evolution during Recrystallization in Aluminum

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1. Introduction Crystallographic preferred orientation, commonly known as texture, is present in essentially all man-made and naturally occurring materials. It is important because the anisotropic nature of most material properties leads to anisotropic behavior in textured polycrystals. Our understanding of texture development is well developed for plastic deformation and complete descriptions are now available in textbooks [1] [2]. Annealing can also lead to profound changes in texture [3] especially if recrystallization occurs. Here we adopt the conventional distinction between local rearrangements of dislocations during recovery and long range boundary motion during recrystallization. After many decades of research, texture evolution during recrystallization is understood but only in a semiquantitative fashion. Symptomatic of this state is the ongoing debate over competing theories of oriented nucleation (ON), oriented growth (OG) and orientation pinning (OP). In addition to these ideas, which are discussed in more detail below, some attention has been given to the possibility of variation in the stored energy of plastic deformation as a function of texture component [16] [4]. Thus we can list the main influences on texture development during recrystallization, in order of their impact as determined in this investigation, as follows: (i) The orientation of the nuclei or new grains (ii) The growth rate of the new grains (iii) The location of new grains belonging to a given texture component relative to one another (iv) The stored energy of the grain into which the new grains grow In general it is the combination of these effects that governs the recrystallization texture. The next section deals in detail with the factors influencing texture and the various theories. A recent experimental study of recrystallization in hot-rolled commercial purity aluminum [5],[7] investigated the kinetics as a function of specific texture components in both the deformed and recrystallized regions. This provided an opportunity to evaluate quantitatively both microstructure and texture development in some detail. By modeling the initial deformed state of the material as a three-dimensional digital material and directly simulating the recrystallization process, it was possible to test various ideas concerning the factors that influence recrystallization, including the standard theories mentioned above. The results suggest that all the factors contribute to varying degrees and none of them can be ignored. Note that the related efforts, primarily in Europe have focused on “through-process” modeling where, to conserve effort, a broader set of processes has been modeled with good success but using statistical assumptions on nucleus placement and growth rather than fully microstructural models [8, 9, 10]. The primary aim of the work is to provide a model with predictive capabilities. Sebald et al.[11] and Solas et al.[12] studied the effect of nucleation on recrystallization texture. They both also highlighted the importance of stored energy on the final texture.


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Work presented by Solas et al. on Zinc showed a comparison between initial and the final pole figures as a means of comparison of the various cases studied. Sebald et al. on the other hand studied the effect of various nucleation schemes on the recrystallization texture. The results were presented in form of the ODF for the final recrystallization textures. The focus of most of the recent effort in modeling recrystallization typically uses the final texture as an indicator of the goodness of fit. The work presented here shows that it is productive to compare measured textures with calculated textures during the process of recrystallization. The justification for the extra effort required to model recrystallization at the microstructural level is that it represents an opportunity to verify quantitative understanding as well as faithfully model the entire process of texture evolution during recrystallization, not just the final state. The literature contains various attempts to model recrystallization textures quantitatively. Köhler and Bunge [13], for example, assert that recrystallization textures in aluminum can be modeled using OG. Doherty [14] introduced factors α and β to describe the effects of ON and OG, respectively. Doherty et al. [15] quantified the effect of OP to the extent of showing that their results were consistent with observed cube band spacings. However, none of these represent an effort to construct a microstructural model and explore the importance of each of these factors by quantitative comparison to experimental data. We now review each factor in more detail. 1.1. Factors influencing texture evolution The most popular theories that have emerged in the past few decades favor either OG or ON. The oriented nucleation theory claims that the origin of the recrystallized texture is in the preferred nucleation of preferred nuclei. The oriented growth theory, however, claims that the origin of the final texture lies in preferred growth of particular nuclei from a more or less random initial distribution. Orientation Pinning states that a given component will grow freely until it runs into a similarly oriented region at which point the recrystallization front will slow down because a low angle boundary is formed. Components that are widely spaced (with respect to the final recrystallized grain size) through the thickness of a rolled sheet will experience little orientation pinning, thus tending to dominate the final texture whereas components present at higher volume fractions will be limited in growth.‡ 1.2. Oriented Nucleation It is now widely accepted that in low solute alloys with high stacking fault energy, new grains are generated from the original deformed matrix by subgrain coarsening. The main assertion of the oriented nucleation theory is that the texture developed during recrystallization is determined at the beginning of the process. In effect the texture ‡ This discussion focuses only on FCC metals, although similar considerations apply to BCC metals


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of the nuclei is latent in the deformed microstructure. Some regions in the deformed microstructure are more likely than others to “seed” the new grains, either through locally high stored energy or high orientation gradients. The most important work for support of this argument was reported by Dillamore and Katoh [18]. They calculated the rotation paths of individual grains during the compression of polycrystalline BCC iron. They showed that transition bands with particular orientations have a higher tendency to form nuclei, which was supported by the experimental observation of those orientations being dominant after recrystallization. Similar experiments in copper by Ridha and Hutchinson [19] show the emergence of cube texture from the the regions where they were predicted. Necker demonstrated how the strength of the cube texture depends on rolling strain [21]. He found a strongly non-linear variation with nearly 80% reduction in thickness required to generate appreciable volume fractions of the cube component. Vatne et al. in their recent work [22] have shown that cube nuclei arise from deformed cube bands. They argued that the cube grains are meta-stable during the process of hot deformation and serve as nucleation sites for the new grains. They also concluded that these recrystallizing grains have a size advantage over the other orientations. Inagaki et al. in a recent paper[28] have reported that for high purity Aluminum rolled to high reduction (98%) the formation of the cube nuclei was not restricted to the cube bands. On the contrary they occurred as isolated equiaxed grains. 1.3. Oriented Growth The alternate hypothesis of Oriented growth (OG) was proposed by Barrett in 1940[29]. The theory of oriented growth is centered around the experimental observations of specific misorientation relationships between growing grains and the deformed matrix. Early experiments by Beck [30][31] on artificially nucleated deformed single crystals showed evidence of oriented growth in the form of rapid growth of new grains having a 40◦ misorientation about a < 111 > axis with deformed grains. This has been reinforced by subsequent work by Liebmann [32], Huang [33] and Taheri [34]. There also have been several experiments on polycrystals which favor the oriented growth theory. There is considerable evidence that certain grain boundaries having specific orientation relationships have high mobilities. The most important of these is the Σ7 boundary type, which is close to 40< 111 > type noted above. To varying degrees, this boundary type has been shown to be the most mobile. By contrast the Σ3 boundary, which is a low energy configuration, is relatively immobile especially in the coherent twin (pure twist) configuration. Molecular dynamics simulation in both 2D [42] and 3D [44] showed a maximum for the value of mobility at Σ7 misorientation. 1.4. Other Theories: Orientation Pinning Both the above mentioned theories have notable exceptions and they alone can not explain the wide range of experimental results. Most notably Juul Jensen [36] showed


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that the growth rate for the cube grains was about 1.5 times higher than the rest, however, there was wide range of possible misorientation between the recrystallized grains and the deformed matrix. Imagine a nucleus growing in a deformed matrix. As this nucleus grows, at the expense of surrounding material, its neighborhood changes constantly. As a result it samples many different orientations during its growth and therefore it is unlikely that a special misorientation relationship can be sustained between the growing cube grains and the matrix grains into which it grows. Instead, the growth advantage of the cube is posited to be simply because all other orientations experience more inhibition of growth from impinging on similar orientations thus forming low mobility boundaries. This has lead to the recent theory on the selection of cube texture as a major recrystallization texture component is “orientation pinning” (OP). Proposed by Juul Jensen [36] it has been substantiated by Doherty et al. [15]. This theory relies on presence of low angle grain boundaries (LAGBs) that arise when growing grains that are members of the same texture component impinge on one another (recrystallized or unrecrystallized). The LAGBs thus formed slow the growth rate of the non-cube grains more than cube grains for the following reason. Assuming that the cube nuclei originate from cube bands and that the cube bands are well separated from each other one can deduce that the cube nuclei will impinge on each other less frequently. By contrast, new grains forming in the dominant texture component will impinge on like oriented grains more frequently and hence will be slowed down or blocked. This theory can thus explain the presence of stronger cube texture if the recrystallized grain size is comparable or more than the band spacing in the deformed microstructure. Engler[24], however, showed that “orientation pinning” alone is insufficient to explain the growth of cube component. This conclusion is reinforced by this work. 1.5. Other Theories: Stored Energy One should also not overlook the effect of heterogeneous stored energy, i.e the possibility that spatial variations in the energy (dislocations) stored in the deformed microstructure could influence growth rates and therefore texture development. Godfrey et al. [37] showed that the recrystallization proceeds much faster, 2 orders of magnitude, in a region with higher stored energy. Hence if cube nuclei occur in the region between cube and S bands then we can expect to observe rapid growth of these nuclei since S bands have been shown to have more stored energy [22]. Interestingly enough, variations in stored energy have long been known been known to govern the order in which nuclei appear in the different texture components in BCC metals [17]. 1.6. Other Theories: PSN There is another important mechanism in recrystallization namely that of particle stimulated nucleation (PSN). PSN depends on the concentration of plastic deformation and orientation gradients around coarse constituent particles. Since the material of


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interest was hot rolled, however, such concentrations do not occur rendering the influence of PSN negligible. Therefore PSN was not considered in this work. By contrast, it is well accepted that PSN is a significant factor in recrystallization of cold rolled aluminum alloys[23]-[27]. Lee proposed an alternate theory [20] to explain the evolution of recrystallization textures. The theory contends that the growth of certain favored texture components is due to the directional stress distribution associated with the anisotropic dislocation arrangement. The cube component emerges as the dominant texture component after recrystallization in most FCC rolled materials. The main intention of the work presented here is to have a model which matches as closely as possible the experimental results in all aspects including texture development throughout the entire recrystallization process. Hence it is necessary to identify the important factors. 2. Experimental texture evolution The experimental results used for comparison with the simulations are obtained from a hot rolled AA1050 sample at 375◦ C [7]. Chemical composition of AA1050 is given in table 1. The experimentally observed texture evolution, which was extracted from EBSD maps, was found to be insensitive to temperature. Accordingly, the annealing temperature for comparison was chosen to lie in the center of the investigated set, which comprised 325, 350, 375 and 400◦ C. To optimize the statistics, data from multiple scans, experimental observations, were pooled together. All the scans were obtained from a plane perpendicular to the Rolling direction (RD) to maximize number of grains in each scan and hence to have a more statistically representative dataset. Table 1: Chemical composition of AA1050 (wt.%) Si Fe Cu Mn Mg Zn Ti Al 0.08 0.31 0.003 0.036 0.004 0.009 0.008 99.54 Figure 1 shows the texture evolution as a function of time in terms of volume fractions of components. The texture components at each time step, or observation point, are calculated by binning the orientation data. The bin for a given data point is decided by its disorientation from the specified texture components. Each Euler angle triplet in the scan is converted to quaternions, say q1 . The disorientation between this q1 and each of the specified components qcomp is calculated and the point is binned by the smallest disorientation angle. If none of the disorientations is less than 15◦ then the point is assigned to the rest component. The texture evolution of each component is reported in terms of volume fraction of that particular component at any given time step. For example the volume fraction of


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Figure 1: Evolution of various texture components (experimental observations in Al1050)[5], [6] the cube component at time t = 900 s is 0.37. The volume fraction, f , at any time, t, is calculated by f=

Volume of the component(t) total Volume(t)

The S component ({231}< 124 >) is the dominant component at the initial time as expected in a rolled aluminum sample. This component decreases rapidly at first and then its volume fraction stabilizes and even increases slightly towards the end of recrystallization. Along with S, the other major rolling texture components, namely copper and brass, make up most of the initial texture. There is a small amount, ∼5%, of cube present initially. At later times, however, the recrystallizing cube component ({001} < 100 >)grows to be the dominant texture. Note that the initial cube component is present in the form of thin bands. The spacing of the cube bands was determined to be greater than the final grain size, which suggests that the cube component would not be expected to be subject to orientation pinning (OP). In both the deformed state as well as the completely recrystallized state there was a significant volume fraction of orientations which could not be categorized into any of the known components for Aluminum. This will be referred to as the “rest”, as noted above. The error bars represent the maximum and minimum values observed for each time point.


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3. System Setup Recrystallization was modeled using the Monte Carlo simulation technique. The setup involves generating a representative microstructure, adding nuclei to the microstructure, adding anisotropy to the system (both energy and mobility) and calculating stored energy for each deformed grain. The understanding and accurate representation of these forms the basis of our model. 3.1. Microstructure The microstructure is generated using the Microstructure Builder, the details of which are provided elsewhere[48]. We begin by extracting the geometry and the orientation data from the experimental EBSD maps. Using the geometry data we build a distribution of ellipsoids which will eventually describe the grains in the microstructure [50]. Using a random set of points we make a Voronoi diagram. Each of the Voronoi cells are assigned to the ellipsoids thus make a space filling grain structure. Areas of grain boundaries and volumes of the grains are calculated. Each grain is assigned a crystallographic orientation such that the difference between both the orientation distribution function (ODF) and the misorientation distribution function (MDF) of the assigned orientations and target distribution is minimized. This microstructure is then fed as an input to the Monte Carlo codes to simulate recrystallization. The system energy is calculated using: N N X n n o X 1X H(si ) γ(si , sj ) 1 − δsi sj + E= 2 i j i

(1)

where γ(si , sj ) is the interaction energy and H(si ) is the contribution to the energy due to an external field.The interaction term (between unlike orientations), refers to grain boundary energy and the second term refers in general to external field which in this context means the volume averaged stored energy of deformation. The first summation, on j, is over the nearest neighbors (this is the distance over which the interactions between various lattice spins are considered to be significant). In the simple cubic three dimensional lattice the count extends to the 26, 1st , 2nd and 3rd , nearest neighbors. The second summation is over all the lattice sites in the system. The transition probability P is given by P =

  

γ(si ,sj ) µ(si ,sj ) γmax µmax γ(si ,sj ) µ(si ,sj ) exp γmax µmax

−

∆E (γmax )k T

∆E ≤ 0 ∆E > 0

(2)

where µ(si, sj ) is the mobility of the boundary between grains si and sj and ∆E is the change in energy associated with the transition. The model described in this paper uses Monte-Carlo, Potts model to simulate the process of recrystallization. The governing equations are given by equations 1 and 2. Input to the model in the form of microstructure and initial properties viz.


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Stored energy, Nucleation, Grain boundary mobility and energy are determined from experimental results. The exact methodology to obtain the input values is described in detail below. Equations 1 and 2 also provide us opportunity of introducing these inputs into the model. H(si ) represents the volume averaged stored energy of deformation and is a function of grain orientation. γ(si , sj ) represents the grain boundary energy between two neighboring grains. µ(si , sj ) represents the grain boundary mobility between two neighboring grains. The information about grain shape and size distribution is obtained and added to the microstructure by using Microstructure builder. The placement of nuclei and their orientation is an extremely important parameter. Algorithm for nuclei placement and texture determination is also discussed in detail later. To obtain an accurate match of initial to the measured texture the initial microstructure should at least have a few thousand grains[39]; as discussed below, however, the algorithms limited us to a lower resolution. The experimental observations, as depicted in figures 2(a) and 2(b), indicate that the grains are elongated along the rolling direction and can be approximated as ellipsoids. These ellipsoids have the longest semi-axes length in the rolling direction and the smallest in the normal direction. To have a statistically similar microstructure, an initial set of slightly equiaxed ellipsoids was generated which was then stretched to get the desired grain shape with aspect ratios being close to 15:4:1 in RD, TD and ND respectively. The microstructure generations and the recrystallization simulations were run on single-CPU computers which limited the resolution that was feasible. A microstructure (grain geometry) with slightly more than 800 grains was generated and it was possible to fit the experimentally measured texture and misorientation distribution to acceptable accuracy [48]. The final microstructure was in form of a cubical lattice of size 500×200×100, as depicted in figure 2(c). Each of the lattice points was labeled by the grain number which can serve as the spin number, q, in the Monte Carlo simulations. All the Monte Carlo simulations used a lattice temperature of 0.9, which is judged to be high enough to minimize the effects of lattice anisotropy for these simulations. More details for generating three dimensional microstructures can be found in [50] and [48].

(a) Planar view

(b) Transverse view

(c) Simulated Microstructure

Figure 2: Deformed Microstructure (a) and (b)(experimental observation on the Al-1050 sample) and Simulated Microstructure (c)(output of the Microstructure Builder)


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3.2. Nucleation Once the 3D digital microstructure has been generated, the next step is to add nuclei to the system. Here the assumption is made that the system exhibits site saturated nucleation. This assumption is made because the system being modeled showed site saturation type nucleation [5]. The algorithm can accommodate other nucleation behaviors such as continuous nucleation and dynamic recrystallization. Each nucleus has a fixed size of three voxels. The process of inserting nuclei can be broken down into two different components. i. nuclei placement ii. nuclei orientation These two components can either be treated separately or as a combination depending on the problem at hand. For example simulation of a system with negligible correlation between the nuclei placement and the nuclei texture permits one to use two separate algorithms to model the process of nucleation. In the present case [6, 5], however, the two are highly correlated, so it was necessary to design an algorithm to address both aspects at the same time. The following algorithm for adding nuclei to the microstructure was employed for this study. Note that this study illustrates the strong contribution from measurement of local texture (microtexture characterization). It also illustrates the level of detail required for characterizing the current state of the material for subsequent prediction of recrystallization. Two important assumptions are made in this implementation. The first of these, as already stated, is the site saturation condition. The other assumption is that the nuclei (new grains) occur either at sites which are adjacent to or are on interfaces between two grains. This second assumption is justified in the present case by the experimental observations[6]. 3.2.1. Algorithm We will first present the algorithm and then study it in more detail with the help of an example. 1. Read the data (microstructure, M, and probability matrix, P§);

2. Build a list of sites on grain boundaries, say L; 3. Pick a site from L, say si ;

4. If si is recrystallized remove from L and go to 3;

5. if si has two or more deformed neighbors proceed, else, go to 3; 6. Read the texture of this si , say oi ;

7. Pick a nucleus texture (equivalent to picking a q value), say on ; 8. If P(on , oi ) is greater than 0 accept the assignment remove si from L and go to 10;

9. Else go to 10;

§ The probability matrix P is more properly termed a frequency matrix since we multiply each entry of the probability matrix with the total number of nuclei needed to make the required volume fraction


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10. Repeat above till required number of nuclei generated. The probability matrix is generated from the experimental observations. Since it is impracticable to have a probability matrix associated with every orientation even in the reduced fundamental zone, the orientations were binned into the major texture components. For the present study the components chosen were Cube, Brass, Copper, S and rest (for both deformed as well as recrystallized). Thus the probability matrix P is a 5 × 5 matrix whose (i, j)th entry is the probability of finding a nucleus with orientation close to Ci next to a deformed grain with orientation close to Cj . Ci and Cj both belong to the above mentioned component list. To speed up the (numerical) process of nucleation, each entry in P was multiplied by the total number of nuclei to be added to give, in effect, a frequency matrix. Every time there is a successful assignment we mark off the (i, j)th entry by one. Example: Assume we happen to choose a site, from L, in the deformed grain having an orientation close S; if this site does not have two or more deformed neighbors then we reject this as a possible nucleation site. Conversely, if it meets the criteria, then we randomly choose an orientation for the nucleus from a preexisting list of possible orientations (generated from experimental observations). Let us consider a nucleus orientation corresponding to brass as an example. If the entry in P corresponding to a brass nucleus adjacent to a deformed S grain is less than or equal to 0 then we discard both nucleus and the location and choose the site again. If the entry is positive then we accept this particular location for placing the nucleus with the particular orientation. The entry P(brass, S) is reduced by one and the site is removed from the list of possible sites for nucleation. Table 2 shows the probabilities used in the simulationsk. These values were obtained from a specimen annealed for 30 seconds at 375◦ C. To achieve this, EBSD maps were obtained and analyzed to get the neighborhood information. The probability of a recrystallization nucleus, i, occurring next to a deformed grain, j, Pij is calculated by: Bij Btot where, Bij boundary pixels between i and j and Btot is the total boundary area. To get P, each entry in the matrix is multiplied by the number of nuclei to be included in the simulations. Example, for a microstructure of size 500 × 200 × 100 and having 6% volume fraction of nuclei each entry is be multiplied by 200000¶. In table 2 the columns represent deformed components (that border newly recrystallized grains) and the rows recrystallizing components. The highest values for the probability of nucleation are along the diagonal. This implies that the new grains are more likely to be formed next to the deformed matrix grains with the same orientation. In other words likelihood of forming a Cube oriented nucleus next to Cube deformed Pij =

k The values reported in here are in good agreement with previous observations, for example that recrystallized cube tends to occur more often next to S ¶ Assuming that each nuclei has a volume of 3 voxels hence the total number of nuclei is given by: Nnuclei = Vtotal × fraction / 3


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Table 2: Probability matrix used to determine nucleus orientation and environment in the simulations Cube Brass Copper S Rest Cube 0.344 0.0515 0.0454 0.219 0.296 Brass 0.0727 0.355 0.0417 0.298 0.232 Copper 0.0155 0.0636 0.324 0.184 0.388 S 0.156 0.0987 0.102 0.353 0.274 Rest 0.122 0.0753 0.0548 0.203 0.522 grain is highest and the probability of forming a S nucleus next to S deformed grain is highest and so on and so forth. In the column corresponding to deformed S component all the cells have higher numbers for probability than other cells. That is to say, the probability of forming a Cube nucleus next to S deformed grain is higher than forming a Cube nucleus next to Copper or Brass. This implies that the neighborhood of a deformed S grain is more likely to generate nuclei than other places (texture components) in the microstructure. This is also in accordance with S deformed grains having the higher stored energy, which is needed for formation of nucleus. 3.3. Energy-Mobility The anisotropy in the system is expressed via anisotropy in energy and anisotropy in mobility. In this simulation scheme, only the sites adjacent to an interface of two grains are allowed to change state (flip spin). Hence only the grain boundary energy and mobility are important. 3.3.1. Energy Anisotropy The energy anisotropy is characterized by the misorientation between the two neighboring grains. We divide the boundaries into low angle boundaries (disorientation θ ≤ 15◦ ) and high angle boundaries (θ > 15◦ ). The energy of grain boundaries in the low angle regime is expressed by equation 3. γ = γ ∗ θ∗ (1 − ln(θ∗ )

(3)

where θ∗ is 15θ◦ and γ ∗ is a normalizing factor. For the high angle boundaries the energy associated is constant at γ ∗ . The only exceptions are certain “special” boundaries. There has been considerable experimental work to investigate the exact nature of the grain boundary energy curve as function of misorientation as well as the full five parameter description (misorientation and boundary normal). For this study, though,only the three-parameter misorientation is considered although this is a more complete parameterization than the more typical (one-parameter) misorientation angle used in much of the literature. Note that the energy and mobility for each possible boundary in the system is computed separately and stored in a look-up table at the beginning of the simulation. Figure 3(a) shows the variation of grain boundary

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Modeling Texture Evolution during Recrystallization in Aluminum 1.2

1.2 grain boundary mobility

1

1

0.8

0.8

Mobility

*

Energy(γ/γ )

grain boundary energy

0.6

0.6

0.4

0.4

0.2

0.2

0

0 0

10

20

30

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misorientation(θ)

(a) Grain boundary energy

60

70

0

10

20

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misorientation(θ)

(b) Grain boundary Mobility

Figure 3: Grain boundary Energy and Mobility as a function of misorientation angle. The cusps and peaks in the energy and mobility functions, respectively, are functions of the misorientation axis in addition to the angle. energy as a function of misorientation angle (θ). The value of energy increases as the misorientation angle increases up to the transition angle value (∼ 15◦ ). The cusps in the energy value shown in the figure are associated with only special boundaries and not all boundaries having those misorientation angles. The special boundaries considered here are Σ3, Σ7, Σ13b and Σ19b. These CSL boundaries have been reported in both experimental work [38] and MD simulations [43][44] to have lower energies. 3.3.2. Mobility Anisotropy As for the energy function, the mobility anisotropy is also characterized by the grain boundary misorientation. Unlike the grain boundary energy which has cusps, or valleys for the special boundary configurations, the mobility function has a low mobility for LAGBs with a sharp transition [45] to high angle grain boundaries. Rising above the general level of HAGB mobility, there are peaks corresponding to the highly mobile boundary types at specific misorientations. Figure 3(b) shows the plot of mobility as a function of the misorientation angle. The peaks in plot correspond to Σ37c , Σ7 and Σ19b boundaries. These CSL values have been noted to have higher mobilities in both experimental and atomistic simulation results [40][42][45][41][46][34]. Note all these are in the < 111 > misorientation axis series. Thus the peaks shown in figure 3(b) are unique points in the 3 dimensional misorientation space. 3.4. Stored Energy Each grain in the system is assigned a single value for the stored energy. It is assumed that there is no local variation of stored energy inside each grain. Further, all the grains having a similar orientation are given the same stored energy. For example, two grains having an orientation close to copper are assigned the same value for stored


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energy. This assumption, though simplistic, is a good first order approximation. The method for determining the stored energy values associated with each component is discussed elsewhere[35]. Table 3 gives the values of stored energy used for the various components+ ∗ [49].

Table 3: Stored Energy assignment Component Stored Energy Vatne values [22] Cube 1.0 1.0 Brass 1.23 0.84 Copper 1.24 1.18 S 1.27 1.21 rest 0.96 1.0

All the rolling components (Brass, Copper and S) have a higher stored energy as compared to Cube. These data agree well with those obtained by Vatne et al. [22]♯ for a similar system. 4. Simulation Conditions Once the microstructure is generated, orientations have been assigned to grains (by the texture assignment process), and nuclei, grain boundary properties and the stored energy have been specified, Monte Carlo simulations can be run. This study examined the effect of oriented nucleation, energy and mobility anisotropy and stored energy on texture evolution. Since the objective of the study is to identify the relative importance of the above mentioned factors, the simulations examined the effect of turning each of the factors on or off. The table 4 summarizes the different conditions for which the simulations were done. An × in a column implies a factor being turned OFF (its effect √ not included). A in a column indicates that a factor being turned ON. 5. Results The results are presented as plots depicting texture evolution of two of the components (Cube and S). These were chosen since the S is the dominant deformation texture component and Cube is the dominant texture component after the recrystallization is complete. Each of the plots shows a comparison between the experimental and the simulation results. The error bars on the experimental results are based on compiling +

All values relative to the cube component ∗ Same stored energy multiplier of 10 was used for all the simulations ¯ δ¯ as stored energy estimator instead of the Read Shockley ♯ The values published in the paper used θ/ relationship

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Oriented Nucleation × √ √ √ √

Table 4: Simulation Conditions Mobility Energy Stored Anisotropy Anisotropy Energy √ √ √ √ × × √ √ × √ √ × √ √ √

Fig. number 4 5 6 7 8

the data from various scan areas (using the max/min values), while the error bars for the simulations are from different runs (typically 3-4 runs per set of conditions and using standard deviations). The initial volume fractions of the S component 0.7 Simulated Cube 0.6

Simulated S Experimental Cube

Volume fraction

0.5

Experimental S

0.4

0.3

0.2

0.1

0 0

50000

100000

150000

200000

Time (MCS)

Figure 4: Random Nucleation shows a discrepancy between the experimental and the simulated values. There are two possible explanations. First, as stated earlier, one needs a few thousand grains to completely describe the orientation/misorientation space whereas the simulated structure has approximately 800 grains. Second, the orientation assignment uses both the ODF and the MDF in the objective function. The component volume fractions depend only on the ODF and so the observed mismatch can be compensated by a better match in the MDF, thereby reducing the overall error in the system in the fitting process. Figure 4 shows the result of drawing the nuclei from a random distribution. The nuclei placement and the nuclei texture assignment were carried out randomly. Stored energy

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and anisotropy effects were included. The simulated texture development is sharply different from the experimental one. Figure 5 shows the result of turning the oriented nucleation and the mobility anisotropy ON. Both Cube and S component are in good agreement with the values at the end of recrystallization but the rate of fall of S is slower than the experimentally observed rate. Figure 6 shows the result of turning the oriented nucleation, stored energy and the mobility anisotropy ON. In this case, the agreement between the experimental and simulated values for the S component is good but the agreement for the cube component remains poor. Figure 7 shows the result of turning the oriented nucleation and both mobility and energy anisotropy ON. In this case, the agreement improves for the cube but slightly worsens for the S. Figure 8 shows the result of turning ON all the factors. This case resulted in the best agreement for both components during the course of recrystallization. 0.7 Simulated Cube 0.6


Volume fraction

0.5

Experimental S

0.4

0.3

0.2

0.1

0 0

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Figure 5: Effect of Mobility Anisotropy

6. Discussion Figure 9 shows the texture evolution of all the relevant texture components for the case which includes the effect of all the factors. Most of the components show a steady decrease in the volume fraction as the time increases. Only “cube”, which is the dominant component at the end of recrystallization, and “Goss” show an increase volume fraction as the time increase. Overall, the texture evolution of the various components in the simulations follow the experimentally measured trends. One objective of the study was to determine the relative importance of the parameters that are know to affect the

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process of recrystallization. Table 4 lists the different combinations of effects used to simulate microstructure and texture evolution. The results are presented in the texture evolution plots in figures 4 to 8. Each of the plots show just two components, namely the dominant deformation component, S, and the dominant recrystallized component, Cube. 0.7 Simulated Cube 0.6


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Figure 6: Effect of Mobility Anisotropy and Stored Energy ON 0.7 Simulated Cube 0.6


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Figure 7: Oriented nucleation and Total Anisotropy

200000

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The results are presented as the time evolution of the volume fraction of components rather than the error between the simulated and experimental values because only six experimental observation times are available. Given the small number of observations, it was easier to evaluate the goodness of fit on the graph than by attempting to compute 0.7 Simulated Cube 0.6


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Figure 8: Oriented Nucleation, Total Anisotropy and Stored Energy 0.7 Experimental Cube

Simulated Goss

Experimental Brass

Simulated Brass

Experimental Copper

Simulated Copper

Experimental S

Simulated S

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Experimental Goss

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Figure 9: Comparing texture evolution of all the relevant components


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an error between time-dependent quantities. We first present the factors in the order of their importance and then consider the rationale behind this ranking. 1. Oriented nucleation 2. Mobility Anisotropy 3. Stored Energy 4. Energy Anisotropy When the nuclei orientations are selected randomly ††, even though Cube is the dominant component at the end of recrystallization, the final volume fraction is still far away from the experimentally observed value between 40-50%. When the results, shown in figure 4, are compared with the results shown in figure 8, where all the other conditions are identical and the nuclei orientations are selected according to the earlier mentioned scheme, it can be concluded that oriented nucleation plays a very important role. Table 5 shows the initial volume fractions of the different components in the case of random nucleation and the experimentally observed distribution. Table 5: Comparison of initial texture for the cases of Oriented Nucleation and Random Nucleation (all numbers reported as percentage of the total volume) Component Experimental Random Nucleation Cube 24 13 Brass 2.4 3.4 Copper 5.3 5.2 S 24 6.0 Oriented nucleation is the most important factor but by no means the only factor required, as is evident by examining the evolution for the other cases. It is fairly clear that a combination of other factors is necessary if not all. Including energy anisotropy, stored energy and mobility anisotropy separately with oriented nucleation, can not provide the desired result. In all the three cases, the cube component increases during recrystallization but either it is not the dominant component and/or the S component does not follow the experimental pattern. For the case of including only the stored energy, the Cube component is dominant but at late times the S component increases again. For the case of including only the energy anisotropy, S never decreases. Including only the mobility anisotropy gives the closest result but the S component does not fall fast enough. Thus one can conclude that the next most important factor governing the texture evolution is the mobility anisotropy. If stored energy and the anisotropy due to mobility are included then the texture of the final microstructure has the best match with the experimental results. †† The texture of each nucleus is chosen at random from the 90 × 90 × 90 Euler space


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6.1. Factors influencing Texture Evolution As noted in section 1.1 the factors controlling texture evolution are: • Oriented Nucleation • Oriented Growth • Orientation Pinning • Stored Energy

One of the aims of this study was to identify the relative importance of these factors which in turn will help us in designing a model to describe texture evolution. To achieve this objective, Monte Carlo simulations were performed while varying the system parameters (factors) stated above. The Oriented Growth theory is centered around the specific misorientation relationships that may exist between different grains. This is due to either lower grain boundary energy and/or higher mobility associated with special boundaries. In fact the anisotropy of grain boundary properties contributes to both OG directly, through the mobility function and to OP as a consequence of the low (energy and) mobility of LAGBs. Hence to gain a better understanding, this study treated the mobility and energy anisotropies separately. Thus we have the following factors oriented nucleation, energy anisotropy, mobility anisotropy and stored energy. Thus the factors in the order of their relative importance are oriented nucleation, mobility anisotropy, stored energy and grain boundary energy anisotropy. The initial rise of Cube and the fall of the S component can be attributed to the effect of stored energy which can be inferred by comparing figures 5 and 6. The difference between these two figures is that figure 6 also includes the effect of stored energy in addition to the oriented nucleation and mobility anisotropy. If one were to compare only the final texture, i.e. at the end of recrystallization, for the cases of including only mobility anisotropy, figure 5, and including all the factors, 8 then the volume fraction of the various texture components are in good agreement. A different way to look at the same data is to compare the ODFs of the microstructures at the end of recrystallization. Figure 10 shows such a comparison. Figures 10(a) and 10(b) depict the final ODFs the case that includes the effect of all factors and the case which includes only mobility respectively. Figure 10(c) shows the difference ODF of the two cases mentioned above. Apart from a few small regions in the Euler space the final texture in the above mentioned two cases is identical. If we were use only the final texture of the microstructure as a measure of similarity to the measured microstructure then we could ignore the contributions from GB-Energy anisotropy and Stored Energy. Comparison of figures 5 and 8 that show the texture evolution, show that, even though the final texture in both cases is similar (at least in the two components shown), the texture at intermediate times is not in good agreement with the experimental results. The case which includes the effects from all the factors is a better match to the experimentally observed texture during the entire recrystallization process. This demonstrates the need to include all the factors as well as the need to use the full texture evolution to determine which effects need to be included in the model

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The stored energy of all the nuclei is the same “zero” which is in accordance with the fact that the new grains are relatively dislocation free and hence have lower stored energy. The stored energy of the deformed grains is determined by the orientation of the grains and the values shown in table 3. Hence stored energy will dominate the early part of the texture evolution since all the nuclei are surrounded by deformed grains. Once the nuclei impinge on each other, however, the anisotropy starts playing an important role. As noted earlier it appears that including the energy anisotropy seems to help the growth of the S component at later times. Figure 3(a) shows the energy anisotropy as a function of disorientation angle. A general high angle boundary has high mobility and also high energy other than the special CSL boundaries. The energy anisotropy has cusps for the special boundaries while the mobility anisotropy has peaks. The cube component is related to S by a misorientation of 40◦ about the < 111 > axis, which happens to be close to the Σ7 boundary type. The mobility curve has a peak at the Σ7 position (figure 3(b)) corresponding to common experimental observation that this boundary is highly mobile . Thus the anisotropy in mobility can help the growth of the cube component when the cube and the S recrystallizing grains impinge on each other. On the other hand the Σ7 position in the energy curve has a cusp implying that the boundary has lower energy. Hence if the effect of energy anisotropy is included and if the cube and S nuclei impinge, then the boundary will be sessile and the cube component can not grow at the expense of S. Hence it appears that overall growth of S component is aided by the energy anisotropy.

(a)

(b)

(c)

Figure 10: Comparison of final ODFs of different cases (a) Final ODF of the case with all factors turned ON (b) Final ODF of the case with only mobility turned ON and (c) Difference between final ODFs of the case with all factors turned ON and only mobility turned ON


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7. Summary and Future Work The combined effect of oriented nucleation, stored energy and anisotropy is required to get the best possible agreement with the experimental results for texture evolution. Orientation dependent stored energy gives the correct behavior of the texture evolution for the initial times when the nuclei are growing freely without impinging on each other. The anisotropy gives the correct behavior for texture evolution at later times when the nuclei impinge on each other. Thus the following parameters all affect the microstructural evolution to varying degrees. • Oriented nucleation • Mobility anisotropy • Stored energy

• Energy anisotropy The new and useful conclusion is that comparison of computed texture development throughout the recrystallization process allows a ranking of the relative importance of the factors to be arrived at. Of course, this ranking applies, strictly speaking, only to this particular aluminum alloy subjected to warm rolling and with certain assumed grain boundary properties. Nevertheless, it points to the possibility of repeating this analysis in other materials in order to ascertain the generality or otherwise of the particular ranking of factors and whether they vary in importance with processing conditions in a systematic fashion. Thus we have addressed the objectives for the work and have been able to rank the effect of each parameter on the overall behavior of the system. The work has also helped to give some direction to the experiments by demonstrating the importance of the determination of the neighborhood of the nuclei and the stored energy in each component. It is acknowledged that, given sufficient computing resources, it would be possible to increase the resolution of the simulations to evaluate gradients, or rate-of-change of agreement between simulation and experiment. However, several factors impede this approach. One is that it is not obvious what the best comparison criterion might be for time-dependent data. Also, although the effect of, say, mobility anisotropy has been presented in a binarized form (on or off), the anisotropy is know to vary with temperature and composition [47]. In principle, therefore, the sensitivity should be investigated with respect to variations in mobility as a function of grain boundary character. Even when the dependence on the boundary normal is neglected, however, grain boundary character requires three parameters to specify a disorientation. Thus it appeared impracticable to approach the problem of texture development in recrystallization as a conventional sensitivity analysis. The hope is that this model will have some predictive power, although it requires direct measurement of the effect of oriented nucleation (and possibly stored energy variations). Also grain boundary properties would have to be known for the alloy and temperatures of interest.


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8. Acknowledgments The authors would like to thank Pennsylvania Technology Investment Authority (PTIA), Alcoa Technical Center and the MRSEC program of the National Science Foundation under Award Number DMR-0520425 for the financial support for this work. We would also like to thank David Saylor for his valuable input to the work. Finally, the interactions made possible by the Computational Materials Science Network (CMSN) under DOE-OBES were very helpful for grain boundary properties. References [1] Kocks, U. F., C. Tomé and H.-R. Wenk, Eds. (1998). Texture and Anisotropy, Cambridge University Press, Cambridge, UK. [2] AS Khan and S. Huang, Continuum Theory of Plasticity J. Wiley & Sons, New York, 1995 [3] Humphreys, F. J. and M. Hatherly (1996). Recrystallization and related annealing phenomena. Oxford, UK, Elsevier Science, Oxford, UK. [4] A. Borbely, J. H. Driver and T. Ungar, Acta mater. 48 p 2005 (2000). [5] M.H. Alvi, PhD thesis, Carnegie Mellon University (2005). [6] M. H. Alvi, S. Cheong, H. Weiland, A. D. Rollett, Materials Science Forum 467-470 p 357 (2004). [7] M. H. Alvi, S. Cheong, H. Weiland, A. D. Rollett, Acta Mater. 56 p 3098 (2008). [8] M. Crumbach, M. Goerdler and G. Gottstein, Mat. Sci. Forum 467-470 p 617 (2004). [9] M. Crumbach, M. Goerdeler, G. Gottstein, Acta Mater. 54 p 3275 (2006). [10] M. Crumbach, M. Goerdeler, G. Gottstein, Acta Mater. 54 p 3291 (2006). [11] R. Sebald and G. Gottstein, Acta Mater. 50 p 1587 (2002). [12] D. E. Solas, C. N. Tome, O. Engler and H. R. Wenk, Acta mater. 49 p 3791 (2001). [13] U. Köhler and H. J. Bunge, Mater. Sci. Forum 157-162 p 803 (1994). [14] R. D. Doherty, Scripta Metall. 19 p 927 (1985). [15] R. D. Doherty, L-C. Chen and I. Samajdar, Mat. Sci. and Eng. A257 p 18 (1998). [16] N. Rajmohan, Y. Hayakawa, J. Szpunar and J. Root (1997) Physica C 241-243 p 1225 (1997). [17] W.B. Hutchinson, Metal Sci. 8 p 185 (1974). [18] I.L. Dillamore and H. Katoh, Met. Sci. 8 p 151 (1974). [19] A. A. Ridha and W. B. Hutchinson, Acta metall. 30 p 1929 (1982). [20] D.N. Lee, Scripta Metallurgica et Materialia 32 p 1689 (1995) [21] C.T. Necker, PhD thesis, Drexel University (1997). [22] H. E. Vatne, R. Sahani and E. Nes, Acta mater. 44 p 4447 (1996). [23] O. Engler, Textures and Microstructures 32 p 197 (1997). [24] O. Engler, Acta mater 46 p 1555 (1998). [25] O. Engler, G. Gottstein, et al. Materials Science Forum 157-162 p 259 (1994). [26] O. Engler, J. Hircsh et al. Materials Science Forum 157-162 p 673 (1994). [27] O. Engler, P. Wagner, et al. Scripta Metall. 28 p 1317 (1993). [28] H. Inagaki and A Umezawa, 14th International Conference on Textures of Materials p 1273 (2005). [29] C. S. Barrett, Trans. Metall. Soc. A.I.M.E. 137 p 128 (1940). [30] P. A. Beck, Advances in Physics 3 p 245 (1954). [31] P. A. Beck and H. Hu, Recrystallization, Grain Growth and Textures ed Margolin, ASM p 393 (1966). [32] B. Liebmann, K. L¨ ucke and G. Masing, Zeitschrift der Metallkunde 47 p 57 (1956). [33] Y. Huang and F.J. Humphreys, Recrystallization and Grain Growth, Aachen, Germany, Springer, p 409 (2001).


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