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Control of the material microstructure in terms of the grain size is a key component in tailoring material properties of metals and alloys and in creating functionally graded materials. To exert this control, reliable and efficient modeling and simulation of the recrystallization process whereby the grain size evolves is vital. The present contribution is a review paper, summarizing the current status of various approaches to modeling grain refinement due to recrystallization. The underlying mechanisms of recrystallization are briefly recollected and different simulation methods are discussed. Analytical and empirical models, continuum mechanical models and discrete methods as well as phase field, vertex and level set models of recrystallization will be considered. Such numerical methods have been reviewed previously, but with the present focus on recrystallization modeling and with a rapidly increasing amount of related publications, an updated review is called for. Advantages and disadvantages of the different methods are discussed in terms of applicability, underlying assumptions, physical relevance, implementation issues and computational efficiency.

The macroscopic behavior of metallic materials is to a large extent controlled by the size and shape of the grains that constitute the material microstructure. The microlevel grain structure will influence macroscopic material properties such as mechanical strength, electrical conductivity, wear and corrosion resistance, ductility, hardness and fatigue resistance. Being able to predict and control the morphology of this microstructure during different metal working processes thus allows the development of tailored material properties, optimized products and more efficient production processes. Understanding and manipulating the material microstructure are key components in the production of functionally graded materials, having engineered properties in different regions.

Fine-grained materials, with grain sizes down to the nanoscale, are becoming increasingly important in many applications, e.g., in the miniaturization of products such as micro-electro-mechanical components (MEMS), in biomedical devices and also in the production of thin metallic films and foils. As one or more physical dimensions of the product are reduced, the microstructure has to be tailored correspondingly in order to maintain required material properties and reliable operation of the product. Recrystallization and grain size control is also of primary interest in the development of high strength steels. From these observations it is clear that grain size and recrystallization are fundamental concepts in materials science and in materials design.

Recognizing that grain refinement through recrystallization can be achieved by exposing the material to severe plastic deformation, several processes such as equal channel angular pressing (ECAP), asymmetric rolling (ASR), accumulated roll bonding (ARB) and high pressure torsion (HPT) have been devised. In order to optimize such processes and to gain further insight into the mechanics of recrystallization, physically motivated and computationally efficient simulation models are vital. Simulations can be used to predict the microstructure evolution, e.g., in terms of grain size and relative grain misorientation, during plastic deformation and also to indicate suitable settings of process parameters such as deformation magnitude, deformation rate and processing temperature. In addition, crystallographic texture, kinetics of grain boundary migration and the size and distribution of second-phase precipitates can be studied through simulation.

The existence and importance of recrystallization as a metallurgical process has been recognized for many years and simulation models of the process continue to evolve and new techniques continuously emerge. This is of course due to the increased knowledge of the physics behind the process but also due to the increasing availability of efficient computer resources. Review papers considering various approaches to recrystallization modeling have been published previously. In [

The present paper aims at providing an updated review of various approaches to modeling and simulation of recrystallization. Continuum mechanical models, cellular automata and Monte Carlo Potts methods as well as more recent methods such as phase field formulations, vertex and level set models are considered. Advantages and disadvantages of the different approaches are discussed. Although the present paper focuses on numerical models, some attention is also given to analytical and empirical models and classical descriptions of recrystallization. Combined formulations such as crystal plasticity/cellular automata models and multi-level simulations using, e.g., cellular automata with finite elements or crystal plasticity models connected to phase field formulations are not considered here since this is beyond the scope of this review paper. But in passing it is recognized that such multi-level simulations provide promising tools to connect macroscopic structural behavior with a detailed description of the microstructural evolution in a material during deformation. Nor is anything discussed on molecular dynamics simulations which have been used to some extent in the study of the details of separate processes during recrystallization. Several of the herein considered approaches are mainly used to model the material behavior on the grain size level. These models can in many cases be employed in simulations using representative volume elements from which the macroscopic material behavior can be estimated through averaging of quantities defining the microstructure. Such homogenization procedures are not covered here.

This review paper begins in Section 2 with a brief discussion on the recrystallization process itself. This is by no means an in-depth description of the complex recrystallization phenomena but highlights the essential features to be captured in modeling of the process. In Section 3, classical and empirical approaches to modeling of recrystallization are mentioned and fundamental results related to the nucleation of recrystallization grains, to grain growth and to grain boundary kinetics are given. Section 4 discusses continuum mechanical models. Monte Carlo Potts models are the topic of Section 5 and cellular automata of Section 6. Section 7 outlines some fundamental features of phase field formulations and Section 8 gives some notes on vertex, or front-tracking, models. Level set models are the topic of Section 9. Some concluding remarks are given in Section 10. Each of the topics mentioned represent active fields of research and in-depth discussions on each field are beyond the scope of this review article where central concepts and especially applications to recrystallization are in focus. References for further reading are given throughout the text.

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. With progressing plastic deformation, the material becomes increasingly thermodynamically unstable. A number of different processes are now active to reduce the stored energy. One of these processes is recrystallization, whereby new grains of relatively low stored energy are nucleated in the microstructure. These grain nuclei can under proper conditions grow to consume the high-energy microstructure in their surroundings, created by the macroscopic plastic deformation. Reducing the internally stored energy, the material is by recrystallization returned to a thermodynamically more favorable state.

Recrystallization is generally accepted to be defined as the formation of a new grain structure in a plastically deformed material. This recrystallization occurs through the formation and migration of high-angle boundaries,

Recrystallization can take place as a relatively slow and temperature-driven process, subsequent to deformation, known as static recrystallization (SRX). Alternatively, during plastic deformation of the material, dynamic recrystallization (DRX) can take place [

Once new grains have emerged, through either DDRX or CDRX, they may grow by grain boundary migration. A driving pressure acts on the grain boundaries due to the jumps in stored energy across the boundaries, allowing the new grains to expand. As grains grow, the grain boundary area and the related grain boundary energy will increase. This acts to lower the grain boundary migration rate and to reduce the grain size. The grain boundary migration kinetics are further complicated in the presence of particles which can give rise to drag forces on the boundaries and partially pin them down. This particle pinning will restrict the progression of recrystallization while, in contrast, particles (of larger size) also may be favorable to the recrystallization process through particle stimulated nucleation, as mentioned.

Some characteristics of the material behavior during recrystallization are worth noting since these should be anticipated in the results obtained from modeling and simulation of the recrystallization process. A first such characteristic is noted as new grains are nucleated at sites of high dislocation density during DDRX. This primarily occurs along grain boundaries whereby it is common to observe “necklace” patterns of recrystallized grains along these boundaries. Second, if a two-dimensional microstructure with homogeneously distributed stored energy is studied, and particle pinning effects are absent, grain growth will be purely curvature-driven. The evolution of the grain structure will be controlled solely by the minimization of interfacial energy. In this arrangement, the equilibrium state will consist of grains separated by boundaries connected at triple junctions with a 120° division between the boundaries. Finally, as new grains of lower dislocation density consume the microstructure deformed through plastic slip, the macroscopic flow stress behavior of the material will be altered. Caused by dynamic recrystallization, flow stress serrations will appear. At lower temperatures or increased strain-rates, cycles of recrystallization overlap and evens the flow stress oscillations. In this case single-peak flow, followed by dynamic softening is observed. At higher temperatures, or lower strain rates, each cycle of recrystallization is allowed to more or less finish before the next one sets in. This results in flow stress oscillations. In both cases the flow stress tends to saturate at some level, corresponding to a relatively stable saturation grain size. This is illustrated in

The process of recrystallization is discussed in depth in the review paper [

The kinetics of recrystallization is classically addressed using the Kolmogorov–Johnson–Mehl–Avrami (KJMA) relation [_{A} are commonly referred to as the Avrami coefficient and the Avrami exponent, respectively. These parameters are related to the nucleation and growth rate during the recrystallization process. It can be noted that _{A}. The Avrami exponent _{A}, given by the slope of the plot, gives some indication of the character of the nucleation process,

As discussed in Section 2, temperature and strain rate are key parameters in recrystallization. Considering these parameters together, empirical models of recrystallization are often based on the Zener-Hollomon parameter, defined as
_{d} a deformation activation energy that is a material parameter. Note that (·̇) is introduced here to denote differentiation with respect to time. The recrystallized grain size _{i}_{i}

As mentioned previously, nucleation of new grains occurs at sites in the microstructure where enough stored energy is present. The complexity of the nucleation process often requires some simplifications to be made during modeling of the event. It is therefore common to see for example grain boundary serration, relative grain boundary motion (shearing and sliding) and twinning mechanisms be disregarded although being vital parts in the nucleation process [

Models that explicitly treat the nucleation event often consider a critical dislocation density _{c}, needed for nucleation to take place [^{2}/2 the dislocation line energy and

Considering a continuous nucleation process, rather than the site-saturated nucleation of the KJMA formulation, the rate of nucleation is conveniently related to the macroscopic effective plastic strain by a relation on the form
_{n}

Once nucleated, the recrystallized grains can grow due to a driving pressure _{C} related to the local grain boundary curvature. This pressure component is derived from the grain boundary energy and is also a function of the local grain boundary curvature. This pressure component act as to restrict grain growth in order to keep the grain boundary energy to a minimum. Additionally, the driving pressure _{D} that is derived from the stored energy and hence related to the jump in dislocation density across the grain boundary. This component drives an expansion of the recrystallized grain into the plastically deformed surroundings. Finally, the driving pressure _{Z} due to the presence of impurity particles which may exert drag forces on migrating grain boundaries or in the limit prevent boundary migration through particle pinning. Considering purely curvature-driven grain boundary migration, the driving pressure is given by

The pressure component due to particle drag appears on general form as
_{V} is the volume fraction of particles with radius _{p}. The grain boundary energy is a function of the crystallographic misorientation _{1} and _{2} as _{1} = 3/2 and _{2} = 1, the original formulation by Zener is retrieved [_{i}

Purely curvature-driven grain boundary migration is considered for example in [

The grain boundary mobility that enters _{m} is the activation energy for grain boundary migration. The pre-exponential term _{0} can be viewed as a function of both temperature and of the crystallographic misorientation across the grain boundary [

The grain microstructure will influence the material behavior on several levels. Microscopically, the evolution of the dislocation density will be influenced by the grain size as grain boundaries pose obstacles to dislocation motion and serve as sites for dislocation accumulation and storage. Macroscopically, this will manifest itself in an influence on the flow stress behavior of the material, for example due to a Hall-Petch component. In addition, the macroscopic strain-rate dependence of the material will be influenced by the grain structure as shown for aluminum in [

In continuum mechanical models of inelastic material behavior, it is common to describe the deformation history by use of internal variables. These are often variables related to the accumulated plastic deformation and the macroscopic deformation hardening of the material. Different aspects of internal variable formulations are discussed in the review papers [_{y} will include a dependence on the average grain size

In addition, during plastic deformation of the material, the increasing dislocation density _{d} will be perceived macroscopically as a deformation hardening. This can be described by

The evolution of the dislocation density will also be influenced by the recrystallization process as the new grains are of considerably lower stored energy than the parent material. Also, when the average grain size decreases, the amount of grain boundary area will increase which results in restricted mobility of the dislocations and dislocation accumulation. The intricate relations between the different physical processes, and across length scales, result in a system of coupled equations that yield the evolution of the internal variables.

In several phenomenological continuum mechanical models of dynamic recrystallization, the critical condition for initiation of recrystallization is taken as a macroscopic critical plastic strain

As the recrystallization criterion is met, the initial average grain size _{0} will be gradually reduced until a saturation grain size _{f} is reached. Frequently the evolution of the average grain size is described by an expression on the form
_{X} and _{X} are parameters that define how fast the recrystallization proceeds with increasing plastic deformation. The McCauley brackets 〈·〉 indicate that no recrystallization will occur until the recrystallization criterion

Another approach to describing the recrystallization process is related to the formulation in [_{V} (_{0},
_{n} is a parameter related to the probability of presence of subgrains large enough to constitute nucleation sites for recrystallization. The recrystallized grain size is finally approximated under the assumption of site-saturated nucleation from a relation on the form
_{d} is a parameter [

Continuum mechanical models where recrystallization is taken into account have the definite advantage of being able to conveniently simulate macroscopic structural behavior influenced by an evolving material microstructure. A further advantage of the continuum mechanical formulations is that they often are readily implemented as material models in existing finite element models. It is however less straight-forward to include microstructure parameters such as grain orientation and re-orientation during the macroscopic deformation process,

The Potts model [

As in the original Ising model, the Monte Carlo Potts algorithm is based on a division of the analysis domain into a grid of _{i}

The total energy _{s} is the stored energy related to the dislocation density at site _{i}_{j}_{i}

The evolution of the grain microstructure is now obtained by employing a Monte Carlo sampling of lattice states. A lattice site is chosen at random and a change in the spin state of the site to another of

Consistent with the Monte Carlo approach, the switch of a spin state is accepted or rejected on account of a switching probability _{switch} (Δ_{switch} (Δ_{0} is the reduced mobility between the neighboring sites with index _{i}_{j}_{s} represents a thermal energy of the simulation analogous, but not directly related, to the physical thermal energy of the system. Note that _{s} is not the physical temperature, but rather a simulation temperature that governs the degree of disorder—or the noise—in the system. Choosing _{s} = 0,

An alternative to

The morphology of the simulated grain microstructure will be influenced by the underlying lattice onto which the grain structure is mapped. The lattice structure will be represented in the results, giving an undesired faceting of the modeled grain boundaries. This may even influence the grain boundary kinetics as the progress of recrystallization can be slowed down or stopped prematurely. Different remedies for this pathology have been suggested. By one approach, the number of neighbor samplings done for each site is increased, usually by considering an extended set of neighboring sites and not only the nearest sites. A second option is to consider other lattice arrays such as changing from a square to a triangular lattice. A third alternative is to set the simulation temperature _{s} > 0 which will serrate the boundaries and by a properly chosen value result in equiaxed grains and correct recrystallization kinetics.

Lacking physical length and time scales, the results from Monte Carlo Potts simulations can be compared to experimental results by different approaches. One option is to relate the length and time scales of the simulation to their physical counterparts and perform a matching of simulated and experimental results. By this method, a real microstructure is mapped onto the simulation lattice, resulting in a matching length scale. In the next step, the simulation is executed until a certain microstructure is achieved that is comparable (statistically) to one obtained from experiments. The simulation time can then be calibrated against the physical time required to reach the same microstructure. This approach is taken in [

Although other approaches to grain-scale modeling of recrystallization have appeared, Monte Carlo Potts models of the recrystallization process are still frequently used. The algorithm is versatile and flexible enough to represent many different physical features and processes. The numerical implementation is straight-forward and decent computational efficiency can be achieved, especially since the algorithm is very suitable for parallelization. Less attractive qualities of the algorithm is the influence of the underlying lattice and the lack of physical length and time scales, although remedies for this have been suggested as mentioned previously.

Monte Carlo Potts models have been used in a vast number of studies on aspects of recrystallization. Some have already been mentioned and others include for example [

Cellular automata are defined on two- or three-dimensional analysis domains and allow for simulation of both spatial and temporal evolution of microstructures. The analysis domain is partitioned into a grid of sub-regions, the cells. The grid is most often defined as regularly spaced although irregular, or random grid, cellular automata have been employed in [

Considering recrystallization, the cell state variables may include the dislocation density, crystallographic orientation and some identification flags, indicating to which grain the cell belongs and if the cell represents recrystallized material or not. If the identification flag of the current cell and any cell in the neighborhood are different, then the current cell is at a grain boundary.

At the beginning of the simulation, state variables are given values to define the initial microstructure. The initial state can be obtained through simulations, e.g., based on crystal plasticity simulations [

The cellular automaton domain can be analyzed under any combination of boundary conditions along the edges including periodic, symmetric and mirror boundary conditions.

Grain boundary kinetics are conveniently described in the cellular automaton using

Considering a fixed grid cellular automaton, the typical cell size _{c} would be the distance traveled by a migrating grain boundary during a single time step. With the grain boundary velocity given by

However, since the driving pressure _{switch} as was defined in _{max} the maximum velocity occurring anywhere in the analysis domain for the present time step. In each solution step, a random number _{switch} the switch is accepted and the current cell is consumed by the approaching grain, otherwise the switch is rejected. This approach prevents the unphysical situation where all mobile grain boundaries advance one cell distance in a common solution step, irrespective of their varying migration rates. Probabilistic cell state switches are discussed in [

If continuous rather than site-saturated nucleation is considered, nucleation of new grains can be incorporated into the cellular automaton algorithm by employing an evolution law for the nucleation on rate form as in

Cellular automata for recrystallization modeling are discussed in the review paper [

As discussed in [

The cellular automaton algorithm offers attractive possibilities in simulation of microstructure processes, one advantage being high spatial resolution. In addition, unlike many other numerical solution schemes and especially those based on continuous fields, cellular automata provide excellent scalability for computer code parallelization, giving computational efficiency Since the microstructure is fully represented, local effects are considered unlike in phenomenological models. Discrete, rather than homogenized, processes are modeled. Cellular automata are also versatile tools in computational materials science since arbitrary constitutive relations and cell state switching rules can be used. They are also capable of replicating the recrystallization kinetics obtained from the KJMA model, as shown in [

Additional examples from the vast number of studies where cellular automata have been used in modeling of recrystallization can be found in [

In phase field 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 correspond 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 in contrast to sharp interface models where jump discontinuities in quantities such as the energy occurs. This is schematically illustrated in

As a consequence of the diffuse interface formulation, there is no need to explicitly trace the location of interfaces as in sharp interface models. This allows arbitrary grain morphologies to be represented without any assumptions on the grain shapes. The evolution of phase field variables in time is calculated from a set of partial differential equations that are solved numerically.

The governing equations for a system of two coexisting phases described by a non-conserved, continuous, phase field _{k}_{k}

_{k}_{i}_{k}_{≠}_{i}_{k}

In addition to the _{k}_{i}_{i}

As discussed initially, the driving force for recrystallization is a minimization of the energy of the system. This energy can be viewed to consist of different components related to interfacial energy, bulk energy, elastic energy and so on. In a phase field setting, the energy is established as a functional of the phase field variables and their gradients in contrast to standard thermodynamics where a homogeneous distribution of the properties is assumed. The approach for systems with diffuse interfaces and heterogeneous properties was first established in [_{k}_{i}_{k}_{k}

Evolution laws for the orientation variables _{k}_{i}_{k}_{i}_{k}_{i}

The evolution laws in

Phase field models have during the last two decades been employed in simulations of a number of different microstructure processes. Arbitrary microstructure geometries such as grains can be represented without the need of explicitly tracing interfaces. Phase field models also provide the possibility to consider a wide array of microstructure processes based on thermodynamic formulations. The computational effort involved in simulating an evolving microstructure using phase fields is quite significant. Remedies such as adaptivity of the discretization grid can reduce the computational time, as can code parallelization although not as efficiently as with discrete methods such as Monte Carlo Potts formulations or cellular automata. Phase field models are capable of tracing arbitrary grain interface geometries and their evolution although the method is less applicable in studies of texture evolution.

Phase field modeling in materials science in general is reviewed in [

Vertex, or front tracking, models for simulation of grain growth have been established by a number of authors, e.g., in [

In vertex models it is common to assume the triple or quadruple junctions to be in equilibrium, although the discussion in [_{i}_{i}

Grain boundary migration is considered as a dissipative process and based on a grain boundary segment of length _{i}_{j}_{ij}_{ij}_{ij}

The summations in

During simulation, the topology of the microstructure changes and transformation rules have to be established. Considering the two-dimensional case, and if only nodes at triple junctions are employed, the transformation rules involve recombination of junctions and removal of three-sided grains below a certain size. These processes as often denoted as _{1} and _{2} transformations. Additional transformation rules are included if also intermediate vertices are introduced [

The velocities of the vertices are obtained from

Vertex, or front tracking, models are mentioned in the reviews [

Vertex models are not as widely used in recrystallization modeling as for example Monte Carlo Potts models and cellular automata, but attractive features of the method include a physical time scale [

Use of level set formulations to model recrystallization is, at least compared to cellular automata and Monte Carlo Potts models, a relatively recent development. The central concept of the method is to trace the position of a moving interface Γ (

The motion of the interface Γ (

Considering a grain microstructure with several grains, the level set formulation is expanded so that each grain _{i}

The level set representation of grain boundaries allows convenient treatment of boundary curvature since it is obtained directly from the level set functions. The unit normal to the boundary is given by

To some extent, the evolution of the level set method resembles that of the phase field method, beginning with systems of two separate phases. The level set formulation was introduced in [

As with the phase field formulation, the level set method also allows a direct representation of interfaces and curvature that is not possible in Monte Carlo Potts models and cellular automata. There is also no need to explicitly treat the interface discretization as is required in vertex models. Fundamental topological changes of the grain microstructure such as the _{1} and _{2} transformations that requires special treatment in the vertex method, as discussed in Section 8, are captured by the level set method without additional considerations. A disadvantage of the level set method is its inability to trace textural evolution. A possible remedy for this might be a combination of crystal plasticity and level set formulations. In [

This paper reviews the main methodologies in modeling and simulation of recrystallization. Analytical and empirical results as well as numerical methods are considered. Numerical formulations that are discussed include continuum mechanical models and discrete approaches such as Monte Carlo Potts models and cellular automata as well as vertex, phase field and level set models.

Both Monte Carlo Potts models and cellular automata are well established in computational materials science and have been widely used in the study of recrystallization phenomenon. They are relatively easily implemented and can be used to capture many aspects of the microstructure physics during recrystallization. High computational efficiency can be obtained using these methods since the discrete nature of the algorithms is well suited for parallelization. Limitations lie mainly in the dependence on the underlying solution grid, representation of grain boundary curvature and in the interpretation of simulation length and time scales.

The vertex, or front tracking, method is a more recent contribution that has been used to some extent in recrystallization simulations. The method is more involved to implement and more computationally demanding. Grain boundaries can be better represented than in Monte Carlo Potts models and cellular automata, but the numerical scheme has to be employed with special consideration regarding topological changes in the microstructure. To include curved grain boundaries, vertices need to be placed between junctions, involving additional calculation steps.

Phase field formulations are being increasingly employed in computational materials science. Most phase field studies on recrystallization have been conducted on grain growth, avoiding the nucleation stage. Such investigations have, however, begun to appear and the development will certainly continue. Phase field simulations are more computationally intensive than for example Monte Carlo Potts models and cellular automata and are also less amenable for parallelization. Still, the representation of surfaces is better and there is no need for explicit tracing of interfaces as in the vertex method. To properly resolve distinct interfaces, adaptivity of the solution mesh or grid is often employed, adding to the computational load.

Level set models of recrystallization are still relatively few although the method has many appealing traits. As in the phase field method, grain boundary migration can be directly established without tracing interfaces. Boundary curvature is conveniently obtained and many aspects of grain structure evolution, such as appearance and disappearance of grains, can be included. As with the phase field method, also the level set representation of narrow interfaces require a fine computational grid or mesh. Again, adaptivity can be used at the expense of extra computation time.

Applying to all of the numerical methods discussed herein, they become computationally more expensive as three-dimensional models and larger numbers of grains are considered.

All of the methods discussed have both merits and disadvantages and selecting what method to use is largely a matter of the physics and complexity of the problem at hand, of available computational resources and of course also of personal preference.

(^{−3}

Illustration of a 2D grain structure mapped onto a square lattice. All lattice sites belonging to a common grain share the same lattice index _{i}

Illustration of a 2D cellular automaton with square cells. Two common types of neighborhood for a cell

Schematic close-up of a grain boundary between two grains

(

Schematic representation of a grain microstructure using the orientation parameters _{k}

Vertex model representation of a triple junction between three boundaries having the interface energies _{1,2,3} and the separation angles _{1,2,3}. The nodes are indicated by circles and the local velocity

_{d}Grain size parameter

_{n}Nucleation parameter

_{X}Exponent in the evolution of the recrystallized grain size

_{0},

_{f}Recrystallized average grain size and its initial and final values, respectively

_{V}Volume fraction of particles

_{X}Coefficient in the evolution of the recrystallized grain size

_{c}Typical cell size (in cellular automata models)

_{0}Grain boundary mobility coefficient

_{A}Avrami exponent

_{V}Density per unit volume of recrystallization nuclei

_{C},

_{D},

_{Z}Grain boundary pressure and components thereof

_{d},

_{n},

_{m}Activation energy for deformation, nucleation and migration, respectively

_{p}Particle radius

_{V}Grain boundary area per unit volume

_{s}Absolute temperature and simulation temperature (in the Monte Carlo Potts model)

_{0}Reduced mobility (in the Monte Carlo Potts model)

_{switch}State switching probability

_{1}

_{2}Parameters related to Zener drag

_{k}

_{d}

_{c}

_{k}

_{y}Macroscopic yield stress