# Nucleation of Recrystallization

As a metallic material is plastically deformed, energy will be accumulated in the material through generation and rearrangement of defects, mainly dislocations. As dislocation networks develop they will tend to form entanglements that subsequently form dislocation cells which, in turn, eventually leads to the formation of subgrains in the grain interiors. These subgrains will be the birthplaces of the recrystallization nuclei. In order for the subgrains to become viable nuclei, however, two criteria need to be met locally in the microstructure. The first is a kinematic criterion requiring mobile high-angle grain boundaries to be formed by the nucleation event. The second is a thermodynamic criterion requiring a stored energy gradient to be present across the interface that is sufficient to provide enough positive driving pressure for grain growth to occur. These criteria are usually found to be met at grain boundary triple junctions and along grain boundaries. Such sites provide enough lattice curvature and sufficient stored energy differences for providing possible nucleation sites. Nucleation may also, but less common, take place in the grain interiors, for example along shear bands and near particle inclusions. The latter is often referred to as particle-stimulated nucleation. As nucleation preferentially takes place along pre-existing grain boundaries, it is common to observe so-called necklace patterns of recrystallized material along the boundaries.

The process of subgrain formation is in mesoscale models usually captured by considering an incubation time $t_{\mathrm{c}}$ or a threshold dislocation density $\rho_{\mathrm{c}}$, i.e. a threshold stored energy, that needs to be reached before nucleation is initiated. In macroscale models, the same process is usually formulated in terms of a critical strain $\varepsilon_{\mathrm{c}}$ that has to be achieved in order for nucleation to take place. Also the notion of a critical stress $\sigma_{\mathrm{c}}$ has been proposed to identify the onset of recrystallization.

During the formation of the subgrains, recovery processes very rapidly reduce the dislocation content in the subgrain interiors by annihilation of dislocations and by accumulation of dislocations at the subgrain boundaries. This increases the subgrain misorientation with respect to neighboring regions and eventually provide high-angle, mobile, boundaries. The same processes also provide subgrains with very low internal stored energy.

The crystallographic orientation of the nuclei will influence both the progression of recrystallization and also the evolution of any recrystallization texture in the material. Recrystallization nuclei are products of the cold worked parent material but they do not necessarily inherit their orientation from the parent grains although it is not uncommon that clear evidence of the texture present in the initial, cold worked, microstructure can be observed also in the recrystallized material. High-angle boundaries with respect to the surrounding material are also required in order to permit growth of the nuclei, causing differences to develop between the initial and recrystallized textures. It can be noted that the development of recrystallization texture is usually attributed to either orientated nucleation or oriented growth or possibly a combination of these processes.

Having defined the initiation of recrystallization, the next question is at what frequency new nucleation events take place. This issue has been approached in a number of ways in different studies. The two main trends is to consider either of  site-saturated nucleation where all nuclei are assumed to be present at the start of the simulation and no new nuclei are added over time or, alternatively, continuous nucleation where new nuclei are continuously added according to some expression of the rate of nucleation. In the latter scenario, both constant and non-constant rates of nucleation have been considered.

Although constant nucleation rates are often assumed, there are experimental studies indicating nucleation to be non-constant and continuous process. However, it is inherently difficult to experimentally determine the rate of nucleation as it is not easy to define through experimental observations if a region in the microstructure is a subgrain, a nucleus or an expanding grain and the exact time at which one changes into the other.

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# Production of very fine grained materials

Since recrystallization is driven by reduction of stored energy, the amount of cold work influences the grain size evolution. Materials with grain sizes down to the nanometer scale can be obtained by exposing the material to severe plastic deformation (SPD). Such SPD processes include Equal Channel Angular Pressing/Extrusion (ECAP/ECAE), Asymmetric Rolling (ASR), Multi-Directional Forging (MDF), High-Pressure Torsion (HPT) and Twist Extrusion (TE). Some of these processes are discussed further below.

During ECAP, studied in Hallberg et al. (2010), the material specimen is pressed through a channel die as illustrated below.

The amount of effective plastic deformation, denoted by $\displaystyle \varepsilon_{\mathrm{eff}}^{\mathrm{p}}$, that is imposed onto the specimen in each pass can be estimated from knowledge of the die geometry according to

$\displaystyle \varepsilon_{\mathrm{eff}}^{\mathrm{p}}=\frac{N_{\mathrm{pass}}}{\sqrt{3}}\left[2\mathrm{cot}\left(\frac{\Phi}{2}+\frac{\Psi}{2}\right)+\Psi\mathrm{cosec}\left(\frac{\Phi}{2}+\frac{\Psi}{2}\right)\right]$

Varying the channel geometry thus allows control of the amount of plastic deformation that is exerted onto the specimen in each pass through the die. If the specimen is rotated between each pass through the die, different standardized processing routes can be obtained.

Another SPD process is asymmetric rolling, discussed in Hallberg (2013), where a conventional rolling process can be made asymmetric by different methods.

The asymmetry of the process can be induced by having different radii $r$ of the rolls, by different roller velocities, i.e. $\omega_{1}\ne\omega_{2}$ or by different friction/lubrication conditions at each side of the sheet. The asymmetry increases the shear deformation of the rolled sheet and hence the total amount of effective plastic deformation.

# Continuum scale modeling of recrystallization

Adopting a continuum mechanical approach, recrystallization can be modeled using an internal variable representation of the pertaining quantities, such as the average grain size and the dislocation density. The macro-scale material behavior will in this way be based on parameters related to the evolving microstructure.

An example of continuum-scale modeling and simulation of ECAP-processing of Aluminum is given in Hallberg et al. (2010). Some results on the distributions of grain size and dislocation density in the work piece are shown in the figures below.

# Recrystallization and grain growth

Recrystallization (RX) is one of the main mechanism to control the evolution of grain microstructures. RX is generally accepted to be defined as the formation of a new grain structure in a cold-worked material and occurs through the formation and migration of high-angle boundaries. The grain boundary migrations are primarily driven by stored energy reduction and minimization of grain boundary surface energy.

As a metallic materials is deformed through plastic slip, energy will be accumulated in the material. This energy is to a large extent expended as heat while the remainder is stored in the material microstructure through the generation and redistribution of imperfections, mainly dislocations. By this process, the material becomes increasingly thermodynamically unstable. During subsequent annealing of the material, reduction of the stored energy can take place through relatively slow recovery or by more rapid static recrystallization (SRX). While the recovery proceeds as a continuous process, SRX is discontinuous. During thermomechanical processing of the material, i.e. when the material is exposed to plastic deformation at elevated temperatures, stored energy generation through dislocation accumulation and stored energy reduction through nucleation of new grains work in parallel. This process is commonly labeled dynamic recrystallization (DRX). The latter process of DRX may be further subdivided into a relatively slow continuous dynamic recrystallization (CDRX) or a more rapidly progressing discontinuous dynamic recrystallization (DDRX).

In materials of high stacking-fault energy, such as aluminum, dynamic recovery is significant and recrystallization occurs mainly by CDRX. In this case, subgrains with low-angle boundaries are formed from dislocation networks. With progressing plastic deformation, misorientation is increased until enough energy is achieved and the initially mobile subgrain walls have become immobilized, allowing new grains to be separated by subgrain growth. In materials of low stacking-fault energy, such as copper, dynamic recovery processes such as cross slip and climb are less influential and the recrystallization is dominated by DDRX during which new grains are nucleated as regions of low dislocation density grow to consume more dislocation-dense surroundings. RX nuclei are commonly accepted to form from subgrains and DDRX will be most significant in the microstructure regions having the highest dislocation density, primarily at grain boundary triple junctions, secondly along grain boundaries and at inclusions and with lesser probability in the grain interiors.

Processing conditions, such as temperature and strain rate, as well as material purity will influence the recrystallization process. This allows some control to be exerted over the resulting microstructure. Simulation models can provide the means for design and processing of materials through recrystallization.

# Mesoscale modeling of recrystallization

Several numerical algorithms have been employed in recrystallization modeling. Some techniques, suitable for mesoscale modeling of recrystallization, are summarized below. A more extensive discussion is given in Hallberg (2011).

#### Monte Carlo Potts models

Historically, Monte Carlo Potts (MCP) algorithms have been used to simulate recrystallization - mainly static recrystallization - in both 2D and 3D on fixed computational grids. The system energy is minimized through probabilistic changes to the state variables, defined at the grid points. Physical time is not available in MCP models where “Monte Carlo steps” are used as a measure of time. This makes comparison between MCP results and experimental results cumbersome.

MCP models are relatively computationally efficient and can be used in both 2D and 3D. The method is also suited for computer parallelization

#### Cellular automata

An alternative method is given by cellular automata (CA) which have been employed frequently in studies on both static and dynamic recrystallization. As with MCP models, CA models are also usually defined on fixed grids, but use physical cell state switching conditions, based on recrystallization kinetics defined in physical time. The cell state switching can be performed as either deterministic or probabilistic. Cellular automata are attractive since high spatial and temporal resolution can be achieved at the grain-scale, cf. Hallberg et al. (2010c) and Hallberg et al. (2014).

The curvature of interfaces is an important aspect of grain boundary migration kinetics but being based on discrete grids with no direct representation of interface curvature, CA algorithms have shortcomings in this respect. The choice of grid type (square, hexagonal etc.) also influences the grain growth kinetics and may be detrimental to the simulation results. As with MCP, CA are computationally efficient and 3D implementations are straight-forward. CA methods also scale well when subject to computer parallelization.

#### Level set models

The level set formulation was introduced as a numerical tool to trace the spatial and temporal evolution of single interfaces. The method was later also extended to consider interfaces with multiple junctions. Standard level set formulations do not correctly capture the interaction between multiple grains, occurring for example at grain boundary triple junctions, and corrections have to be implemented to remedy this shortcoming. Level set modeling of recrystallization is discussed in Hallberg 2013 and of grain growth in Hallberg 2014.

#### Front tracking or vertex methods

Interface migration can also be described by front tracking or vertex models where the migration kinetics of grain boundary triple junctions are considered. The topology of the grain structure is represented by nodes placed at the triple junctions, interconnected by grain boundary line segments. The representation of curvature, however, comes with additional computational cost as intermediate nodes have to be introduced between the triple junctions. In addition, the extension of the method to 3D is not easily realized since it requires surface tesselations to be performed. Also, topological changes to the grain structure require dealing with different transformation conditions. It can also be noted that usual formulations of front tracking models do not hold information related to the grain interiors.

A combined crystal plasticity and vertex model is established in Mellbin et al. 2015, to allow concurrent modeling of large deformations, texture development and recrystallization.

#### Phase field models

The Phase-field method (PF) has received significant interest in recent years in simulations of a broad spectrum of physical processes, including recrystallization. In PF models of recrystallization, the grain microstructure is described by phase field variables. These are functions that are continuous in space and a distinction is made between conserved and non-conserved variables. A conserved variable is typically a measure of the local composition whereas a non-conserved variable contains information on the local structure and could represent for example the crystallographic orientation. Within a single grain, a phase field variable maintains a nearly constant value that corresponds to the properties of that grain. Grain boundaries are represented as interfaces where the value of the phase field variable gradually varies between the values in the neighboring grains on opposing sides of the grain boundary. Grain boundaries are hence described as diffuse transition regions of the phase field variables. The computational effort in treating the rapidly changing fields across diffuse interfaces can be considerable and the formulation of the energy densities to capture physical microstructure features is not trivial. In addition, topological changes such as nucleation of new grains are not easily handled.

# Animation showing a 2D level set simulation of Dynamic Discontinuous Recrystallization (DDRX)

A simple 2D simulation of dynamic discontinuous recrystallization (DDRX) in pure Cu at an elevated temperature. The simulation is based on level sets in a finite element setting. Adaptive remeshing is performed in each step. The animation speed is increased compared to actual time.