Three-dimensional Frontal Cellular Automata Model of Microstructure Evolution – Phase Transformation Module

Abstract

The paper presents three-dimensional frontal cellular automata (FCA) based model for modeling of microstructure evolution during technological processes. It is hierarchical system. The first level is FCA, the second level is modules of microstuctural phenomena; and the third level is models of technological processes. The phase transformation module (PTM) is one of the components of the second level. PTM will contain several models of phase transformation; one of them presents transformation of austenite into ferrite and perlite. This phase transformation controlled by diffusion is considered as the nucleation and the growth of grains of other phases. The nucleation algorithm is presented in the paper. An effect of nucleation sites on final microstructure was studied on three extreme nucleation variants: nucleation on the boundaries, on the edges and in the grain corners. Simulations have been carried out for low cooling rate and relatively long time of the holding at appropriate temperature. The simulation results of the microstructure evolution studies are presented in the paper.

1. Introduction

The prediction of the microstructure and its digital representation in the micro-, meso- and macro-scale is one of the most important problems in materials science. Along with analytical and empirical models, for modeling of microstructure evolution Monte Carlo Potts models,¹⁾ the finite element method (FEM) based models,²⁾ the phase field,^3,4) multi-phase-field⁵⁾ models, the front tracking method^6,7) and the vertex models⁸⁾ are used. However, cellular automata (CA) models⁹⁾ occupy the first place among the methods which represent microstructure and its evolution. The application of the CA models for the simulation of the different phenomena in materials has become incredibly important during the last years. CA are used for modeling of crystallization (solidification),^{10,11,12,13,14)} dynamic and static recrystallization,^{15,16,17,18,19)} phase transformation,^17,20) cracks propagation,²¹⁾ etc.

Generally, material properties are determined by a chemical composition and microstructure. Difficulties in determining material properties are explicitly connected with a lack of models and methods, which would consider a contribution of particular elements of microstructure and could transfer it from the micro scale to the macro one. Additional difficulty lies in a shortage of computational methods, which design a preset structure and study its evolution.

A disadvantage of CA is the fact that it does not give material properties explicitly. But discrete presentation of the structure obtained by CA can be easily transformed it into a form convenient for other numerical calculations, for example, the finite element method (FEM). Such multiscale models are widely used. It should be stressed that microstructure evolution is a complex of definitely three-dimensional (3D) phenomena, only 3D models can properly reflect microstructure evolution and 3D model should be developed.

The most models based on cellular automata, which can be found in literature, are two-dimensional (2D). 2D CA models are the simpler and faster, have less elements and connections, use less complicated algorithms, are simpler for design, implementation, more useful for visualization. But microstructure evolution is pointedly three-dimensional and results obtained by the 2D CA not always can be transferred to a real 3D process. Five problems appear during the 2D CA modeling. They are kinetics of transformation, location of nuclei, grain growth rate, deformation of grains and crystallographic orientation.

Kinetics. Transformation fraction χ can be calculated via extended volume as following:   

$$\chi =1-exp\left(-V_{ex}\right),$$(1)

where V_ex can be expressed as following: V_ex = βtⁿ, β and n are coefficients, which depend on nucleation and grain growth rate.

If we consider a process with nucleation before the grain growth and constant growing rate, extended volume is the cubic dependence on average grain diameter D or grain growing rate v: $V_{ex}=N\frac{\pi D^3}{6}=N\frac{\pi v^3}{6}{t}^3$ . In the 2D CA, it is impossible to receive the same cubic dependence, it is the squarelaw only: $V_{2D}\equiv S=N\frac{\pi v^2}{4}{t}^2$ . If in real process we receive kinetics described by equation with exponent n = 2, it means, the grain growing rate during the process decreases. We obtain the same in 3D CA, but in 2D CA grain growing rate for the same kinetics should be constant. In any case either kinetics or grain growth rate is always improper in 2D CA simulations.

Nucleation. It is well known that the preferred location for the nucleation is on the boundaries (edges, corners) of the grains. The 2D CA is considered as a cross-section. Then, it is very low probability, that nucleon appear in this cross-section. When new grain grows into this cross-section, it probably appears neither on boundary, nor at proper time. No one of known 2D CA models takes that into account, maybe, because dependences of location and time for such a simulation are unknown.

Grain growth rate. If 2D CA considered as a cross-section, then growth rate of grain cross-section will be different from real grain growth rate, especially if nuclei appear outside the considered cross-section. Moreover, if grains grow with constant rate, the growth rate of cross-section must change. No one model introduces such dependence of growth rate.

Deformation of grains. Microstructure after the deformation is different in different cross-section, as some different materials properties in different direction. It cannot be considered in 2D CA.

Crystallographic orientation. 3D crystallographic orientation can be easily taken into account in 2D CA, but not always.

The objective of the paper is a development of the model of transformation of austenite into ferrite and perlite. This model is one of component of the phase transformation module of hierarchical three-dimensional model for modeling of microstructure evolution during technological processes.

2. FCA Based Model

The purpose of the model and the expected results are the driving force for development of the model and its extension. Basing on selected issues, microstructural phenomena and technological processes, which are expected to be simulated, a general block-scheme of the hierarchical model²²⁾ presented in Fig. 1 was developed. The basis for this system is FCA, while the tip is a set of processes. Models of microstructural phenomena are between them and connect them. The materials database completes the system.

Fig. 1.

FCA based model for a simulation of microstructure evolution in technological processes.

2.1. Frontal Cellular Automata, Deformation and Space Reorganization

FCA are a foundation of a modeling system (Fig. 1). FCA are a complex of the millions of cells with the same multi-states automaton (Fig. 2). The set of the automaton states comprises the initial matrix state q₀, the “frontal cell” q₁, the “boundary cell” q₂, the “cell inside the grain” q₃, the transient state q₄ and nucleon state q₅. Two additional states q₆ (“cell in the edge of the grain”) and q₇ (“cell in the corner of the grain”) are introduced to the model of phase transformation. They are described below in the next section. The initial state can be either Q_in = {q₀} for the creation of the initial microstructure, or Q_in = {q₂, q₃, q₆, q₇} for other algorithms. The final state is always Q_f = {q₂, q₃, q₆, q₇}, although, if some stages of the process are not complete, the states q₀, q₁ and q₄ can be presented in the final structure. Frontal cellular automata and its computational advantages are described in detail elsewhere.^18,22,23)

Fig. 2.

Graph of cellular automaton.

Cellular automata play a twofold role (Fig. 1). At first, they serve for the creation of initial microstructure. Then, they are responsible for all microstructure changes during a phase transformation and other phenomena.

The real deformation is the problem that cannot be avoided in CA simulations, especially when the multi-stage deformation is modeled. The shape, sizes of cells and cellular space do not remain the same during the simulation of processes with deformation. A significant cell distortion makes the characteristics of the FCA (isotropy) and quality of obtained results worse. Later, space and cells reorganization should be done. It is realized by the blocks “Deformation” and “Space reorganization”. Consideration of the deformation and space reorganization and their implementation into FCA is described in detail elsewhere.^18,23) Though, they are not applied for the phase transformation, which is modeled without prior deformation, and cannot be concerned in the paper.

FCA use different boundary conditions: open, half-open, periodic and periodic with displacement. The choice of boundary conditions depends on modeled process. Except for the first condition (half-open), all the others can be applied for the phase transformation simulation, although the simplest (periodic) boundary conditions are only used in the paper.

2.2. Microstructural Phenomena and Preparatory Function

Modules of microstructural phenomena are the second level of the modeling system (Fig. 1). Some of them are described elsewhere. Module “Initial microstructure”²⁴⁾ played preparatory and auxiliary roles. It creates a microstructure that can be used in further simulation. Initial microstructure takes into account the shape of the grains, the distribution of their size; crystallographic orientation and a boundary disorientation angle.

Module “Crystallization”¹⁴⁾ is the second block that can create initial microstructure.

Module “Recrystallization” is responsible for a microstructure evolution during the hot forming processes and simulates static, dynamic and metadynamic recrystallization by taking into account evolution of dislocation density, hardening and recovery. Some results can be found in.^18,22,25)

Module “Phase transformation” is created to obtain a microstructure after phase transformation and will contain several models. Model of austenite-ferrite and austenite-perlite transformations is described in this paper. The phase transformation is considered as the nucleation and the growth of grains of other phases. The initial states of the cell before transformation can be arbitrary (q₁ – q₇). The words (I₀∨I₄)I₁(I₂∨I₃) are responsible for the phase transformation in FCA. The process is modeled in consideration with thermal condition, which affects a nucleation rate and a growth rate of ferrite grains. A new variable determined phase is introduced into the automaton. That model is described in detail in the following sections.

The last block of the second level concerns grain refinement in processes of severe plastic deformation.^22,27,28)

2.3. Technological Processes and Materials Database

Another level of layout contains models of technological processes. Basing on the models of microstructural phenomena and putting them in the proper order, they can simulate the microstructure evolution in technological processes. Solidification in continuous casting,¹⁴⁾ hot flat and shape rolling,^18,23,25,29) roll-bonding process²²⁾ and MAXStrain^® technology²²⁾ are among the modeled processes. The last element of the system is the material database.

3. State of Art in Simulation of Phase Transformation – CA Models

The main tools for the modeling of phase transformations are cellular automata, often combined with finite difference method (CA+FDM)^30,31) or finite volume method (CA+FVM),^32,33) as well as phase-field (PF)³⁴⁾ or multi-phase-field method (MPF).³⁵⁾

Probably one of the first attempts of phase transformations modeling was the work of Iba et al.³⁶⁾ By the end of the 20th century models that take into account the actual conditions of transformation more precisely were created. The publication of Kumar et al.¹⁷⁾ could be called a breakthrough. They have taken into account the ferrite nucleation at austenite grain boundaries and used it for the nucleation model developed early by Thevoz et al.^13,37) for solidification (crystallization). The nucleation rate is defined as the Gaussian function with the following parameters: maximum nucleation rate N_max, temperature for maximum nucleation rate T_max and variance of nucleation temperature ΔT_σ   

$$I=\frac{N_{max}}{\mathrm{\Delta }{T}_{\mathrm{\sigma }}\sqrt{2\pi }}exp\left[\frac{{\left(T_{max}-T\right)}^2}{2\mathrm{\Delta }{T}_{\mathrm{\sigma }}^2}\right],$$(2)

Grain growth rate is dependent on carbon diffusion coefficient using solubility balance described by Kurtz and Fischer:³⁸⁾   

$$v=D_{\gamma }\frac{d{C}_{\gamma }}{d\mathbf{n}}\frac{1}{C_{\gamma }^{*}\left(1-k\right)},$$(3)

where dC_γ/dn – concentration gradient of carbon in austenite in the direction perpendicular to the boundary of growing grain; C_γ^* – concentration of carbon in austenite according to phase diagram for the current temperature; k – the distribution coefficient of carbon in austenite and ferrite; D_γ – carbon diffusion coefficient in austenite.

Kurtz and Fisher³⁸⁾ also studied the kinetics of the transformation at a different cooling rate. Varma et al.³⁹⁾ have developed this model to separate nucleation and growth of ferrite and perlite, but the implementation of the cellular automata model seems to be too schematic.

Another very interesting model was proposed by Zhang et al.^40,41) They used iron-carbon phase diagram to determine the nucleation of ferrite grains. The diffusion equation is used for calculation of grain growth, which together with the determination of grain growth rate is resolved outside CA; and stochastic transition rules are applied for the grain growth in CA. According to the model of Umemoto et al.⁴²⁾ during the continuous cooling from the temperature A_e3 to a set temperature T, the number of new ferrite grains n(T) can be determined by the following equation:   

$$n\left(T\right)=\underset{T}{\overset{A_{e3}}{\mathrm{\int }}}\frac{I\left(T^{\prime }\right)\left(1-f\right)}{Q}d{T}^{\prime },$$(4)

where: I(T') – ferrite nucleation rate on the surface of the grain boundary unit at T'; f – ferrite volume fraction; Q – cooling rate.

Ferrite nucleation rate is defined by following equation:   

$$I=\frac{K_1}{\sqrt{k{T}^{\prime }}}{D}_{\gamma }exp\left[-\frac{K_2}{k{T}^{\prime }{\left(\mathrm{\Delta }G\right)}^2}\right],$$(5)

where: K₁ – coefficient responsible for nucleation density; K₂ – austenite-ferrite interphase boundaries energy; k – Boltzmann’s constant; D_γ – diffusion coefficient of carbon in austenite; ΔG – difference of Gibbs’ free energies - the driving force of austenite-ferrite transformation.

Growth rate v of ferrite grains is calculated according to the equation:   

$$D_{\gamma }\frac{\partial C_{\gamma }}{\partial \mathbf{n}}-D_{\alpha }\frac{\partial C_{\alpha }}{\partial \mathbf{n}}=v\left({C^{\prime }}_{\alpha }-{C^{\prime }}_{\gamma }\right),$$(6)

where: C_γ, C_α – carbon concentration in austenite and ferrite respectively; C'_γ, C'_α – carbon concentration near the grain boundary in austenite and ferrite respectively; D_γ, D_α – diffusion coefficient of carbon in austenite and ferrite.

Precipitation of cementite is also considered, although for hypoeutectoid steel presented in the papers^40,41) it seems to be an unnecessary procedure.

Lan et al.⁴³⁾ were the first who introduced a deterministic rule for ferrite grain growth in low carbon steel into a model similar to the model of Zhang et al.^40,41) Later, Lan et al.⁴⁴⁾ took into account the deformation prior to transformation. The grain growth rate is calculated according to the equation:   

$$v=mF,$$(7)

where: m – boundary mobility; F – driving force, which consists of a chemical (F_chem) and deformation (F_def) components.

Next, Zheng et al.^{32,33,45,46,47)} have modeled the “dynamic” austenite-ferrite transformation for low-carbon steel induced by deformation. Li and other⁴⁸⁾ have introduced incubation period for nucleation in accordance with the classical theory of Lange et al.:⁴⁹⁾   

$$I=I_{s}exp\left(-\frac{\tau }{t}\right),$$(8)

where: I_s – nucleation rate according to Eq. (5); t – time; τ – incubation period, which is defined by:⁴⁸⁾   

$$\tau =\frac{12kT{a}^4{\mathrm{\sigma }}_{\alpha \gamma }}{D_{\gamma }{x}_{\gamma }{v}_{\alpha }^2{\left(\mathrm{\Delta }{G}_v\right)}^2},$$(9)

where: a – means lattice parameter for phase of austenite and ferrite; x_γ – mole fraction of carbon in austenite; σ_αγ – interphase energy of disordered ferrite; v_α – volume of iron atoms in ferrite. Values of the parameters can be found in.⁴⁹⁾

Bos et al.⁵⁰⁾ and Mecozzi et al.⁵¹⁾ presented a three-dimensional model of phase transformation of dual phase steel (ferrite + perlite) during annealing, which considers the ferrite recrystallization and austenite-ferrite, perlite-austenite, austenite-ferrite and austenite-martensite transformations.

4. Phase Transformation Model

In the paper the main attention is paid to a diffusional transformation of austenite. Moreover, transformation at a low cooling rate is considered, i.e. austenite transformation to perlite and what proceeds – its ferrite precipitation (austenite-ferrite transformation) for hypoeutectoid steel. The only perlitic transformation without careful examination of cementite precipitation is considered for steel with higher carbon content (hypereutectoid). It is assumed that precipitation takes place at grain boundaries by creating a grid of fine particles of cementite. It does not change the original shape of the grains, but creates some barrier to the growth of perlite grains into different austenite grains. Ferrite precipitation on the austenite grains boundaries also creates such a barrier, but the gaps between the ferrite grains make it possible for a perlite colony to grow into a few austenite grains.

A basis of the model is frontal cellular automata (Fig. 2). Diffusional phase transformation can be described in the same way as other microstructural phenomena based on the nucleation and grains growth, but in this case the grains grow within the grains of other phase.

The initial microstructure for austenite is created by another module (“Initial Microstructure” in Fig. 2). Every grain obtains code “austenite”. Then, neighborhood of every cell is studied. If cells in their neighborhood have cells belonging to other grains, they are set in the state q₂ (one different grain), q₆ (two different grains) or q₇ (three or more different grains). Otherwise the cells are in the state q₃. Then, initial state of the cells for simulations is Q_in = {q₂, q₃, q₆, q₇}.

The nucleation changes the state of the cells. Because nucleation begins in the corners, on the edges and boundaries of the grains only cells in the states q₂, q₆, q₇ are considered as the sites for nucleation. Nucleation conditions for these cells are tested (condition I₀ in Fig. 2) and appropriate cells change their state (q₅ – “nucleon”). For every cell in the state q₅, a new grain is created. Grains contain information about nucleon location, time of nucleation, parental grain, crystallographic orientation, volume, average carbon concentration, code of grain (“ferrite” or “perlite”) and other. Then cell without delay change its state (q₁ – “frontal cell”).

The frontal cells q₁ study their neighborhood. Neighboring cells belonging to austenite grains are attached to the growing grain (ferrite or perlite). It is condition I₄ (Fig. 2). Neighboring cells transits from the states q₂, q₃, q₆, q₇ into the state q₄ (“transient” state). Then, after condition I₄ tested and state changed for all neighbors of all frontal cells, they study their neighborhood again. As results, frontal cells change their state (conditions I₂, I₃ and other in Fig. 2). New state depends on a number of different grains in cell neighborhood (q₂, q₃, q₆, q₇ as described in previous paragraph).

Transition in the state q₄ is accompanied with calculation of time delay (condition I₁). Time delay control grain growth rate (mobility of the grain boundary) and takes into account different factors: temperature, carbon concentration, crystallographic orientation, shape of grains, etc. Time delay determines time, when cells transits into the state q₁. And then calculations repeat.

4.1. Model of Nucleation

Model of nucleation is based on Eqs. (4), (5), (8) and (9), which determine number of nuclei. Nucleation sites are considered in detail below in this section.

Nucleation during the diffusional transformation of austenite, as well as during solidification and recrystallization, is clearly heterogeneous. During solidification, nucleation begins at a super cooled mold surface and can be examined as uniform across the surface, as it cannot be taken into account the curvature of the crystallizer compared to the size of the nuclei, as well as surface roughness is uniform. During recrystallization the heterogeneous nucleation occurs on the grain boundaries, but it is not revealed a preferred nucleation on the edges or in the corners of the grains. While during the austenite transformation, nucleation probability is highly dependent on the location of nucleation sites. Classical Nucleation Theory (CNT)^52,53,54) considers two types of nucleation: homogeneous and heterogeneous. As the parent phase is not unstable, transformation cannot proceed because perturbations in a metastable state increase the free energy. Such perturbations can appear because of thermal. To reach its equilibrium, the system has to overcome an energy barrier so as to form clusters of the new equilibrium phase, a process known as nucleation. When homogeneous nucleation has been considered: it was assumed that nuclei can form anywhere in the system. But it may require less energy for the nuclei to form heterogeneously on preferred nucleation sites. These sites can be at the interface with existing impurities or some lattice defects like grain boundaries or dislocations. The first difference is the cluster size. The nuclei free energy decrease when they are located at a preferred nucleation site. Such a decrease usually arises from a gain in the interface free energy: it is more favorable for the nuclei to form on an already existing interface as the cost to create the interface between the old and the new phases is reduced. Huang and Hillert,⁵⁵⁾ Offerman et al.^56,57) experimentally found that grain corners are the most effective nucleation sites in steel.

As cooling rate and overcooling level is relatively low in further simulations only heterogeneous nucleation is considered. Free energy depends on the amount of old interfaces in neighborhood; the more boundaries are the lower free energy is. So, free energy is lowered in the sequence: inside grain, on convex, plain and concave boundaries, on the edges (from high to low angle between the boundaries), in the corners. Then, the highest nucleation probability is for the corners, then less for the edges and much less for the boundaries. The nucleation probability for the corners and edges depends additionally on the angles between the grains walls that form the edge or the corner.

A significant difference in the probability for different sites introduces another nucleation algorithm and appropriate modification of cellular automaton (Fig. 2). State “on grain boundary” q₂, which was sufficient for the determination of nucleation sites for recrystallization or solidification, when all the probabilities are equal, is not effective for the phase transformation.

According to the assumptions, nucleation probability p_g can be determined as a function of process parameters (temperature, dislocation density, degree of supercooling and others); and its value indicates the number of nuclei appearing on the surface unit (either the edge length or a number of corners) per unit of time. Nucleation can be implemented into algorithm in two ways: locally and globally. Local rules are implemented into most cellular automata models, their principle is based on scanning all possible nucleation sites (each cell), drawing a random number p_i distributed uniformly between 0 and 1, and comparing with the nucleation probability p_g. If p_i ≤ p_g, nucleon occurs in the cell, if not, cell remains in the previous state. From the point of view of computational efforts, it is irrational. In the frontal cellular automata another faster algorithm is used.

The number of nuclei N_n per unit of time can be calculated on the basis of nucleation probability p_g and nucleation surface S: N_n = p_g S or N_n = p_g S_av n_b, where S_av is a surface of the cell face and n_b is a number of cells on grain boundaries (in the state q₂). Actually, the use of the local rules gives the number of grains of the Gaussian distribution with expected value equal to the calculated value N_n. The second, global approach gives the exact value N_n. If, instead of the exact number N_n the Gaussian distribution is applied, the results of both approaches will be the same. However, the second approach does not require scanning, drawing and comparing at each calculation step for each cell. It is sufficient to calculate the number of nuclei N_n and distribute them randomly in possible nucleation sites.

Conditions of nucleation during transformation require changes of algorithm. Two solutions are possible. A solution, which is often used in models not related to CA, can be applied. Then the nucleation conditions are distinguished according to the nucleation sites (boundaries, edges, corners) and they do not take into account the internal angle between the grain boundaries. The second solution considers not only the sites, but also internal angles.

In both approaches, two additional states can be introduced into the FCA, which determine if a cell is located on the edge or in the corner of a grain. Cells in the state q₂ (on the grain boundary) can be used for such additional cell differentiation. If the calculations are a continuation of the previous simulation or the initial structure is read from the file and, therefore, does not provide information on the state of the cells, additional states differentiation are fulfilled simultaneously with determination of two other states (q₂ and q₃). Two additional states are: q₆ – “cell on the grain edge” and q₇ – “cell in the grain corner” (Fig. 2).

Later, three kinds of the cells are formed, in which nuclei can appear; and nucleation probability is different for each kind of the cells. The nucleation algorithm previously used for one type of cell and described above is sufficient to be extended to the other two kinds of cells. At each step, the probability and the corresponding number of cells is determined by the number of nuclei for each cell type and cites for nucleation are chosen randomly from among all the cells of the appropriate type. Probabilities differ not only by value, but also in units of measure. They are probabilities related to units of time and the contact surface of the two grains (p_f), the length of the edge (p_e), or the number of corners (p_n). Therefore, these probabilities are essentially directly incomparable and their proportion can be estimated in a relation to the entire volume of the modeled space, by taking into account the microstructure parameters such as the ratio of the grain surfaces, the length of the edges and the number of corners. The parameters depend mainly on the average grain size. This algorithm deprives tedious counting probabilities for each cell in each step of calculation. It is a basic algorithm and number of nuclei N_n is equal to:   

$$N_n=p_f{n}_f{S}_{av}+p_e{n}_e{l}_{av}+p_n{n}_n,$$(10)

where: p_f, p_e, p_n – probabilities of nucleation respectively on the boundary, on the edge and in the corners of the grains; n_f, n_e, n_n – number of cells on the boundary, on the edge and in the corners; S_av – average area of the cell face; l_av – average length of the cell edge.

The second variant of nucleation, which takes into account the angles, has also been developed. The number of cells in extended neighborhood belonging to the same grain corresponds to the solid angle. It can be calculated for each cell on the grain boundary. The larger neighborhood radius means the longer time calculations. Therefore, radius is limited to two cells (or diameter of neighborhood sphere is five cells). Together with the original cell, such neighborhood consists of 81 cells. 51 cells in the setting correspond to the planar boundary (Ω = 2π), 32 cells to the edge with right dihedral angle (Ω = π) and 20 cells to the corner with three right dihedral angles (Ω = π/2). 35 cells correspond to three equal dihedral angles (Ω = 2π/3) on the edge and 23 cells to four equal solid angles in the corner (Ω = π/2). The difference between the number of cells and the solid angle for edges and corners is an additional difficulty. It appears because the boundary passes not through the center of a cell, but through its face, edge or vertex. That is why, both neighboring cells on the two sides of planar boundary have 51 neighbors of the same grains (not 81/2, and 51 + 51 > 81 cells). The nucleation probability p_i is approximated, depending on the number of appropriate cells in the extended neighborhood. The reference points for approximation were the number of cells of 23, 35 and 51, which are associated with probabilities p_n, p_el_av and p_fS_av, respectively. Additional cells differentiation on the states q₆ and q₇ can be left in this algorithm, but it is not necessary.

If the previous calculation is based on the number of cells and their geometrical parameters, now the number of nuclei N_n can be calculated by aggregating nucleation probability p_i for all cells at the grains boundaries according to the following equation:   

$$N_n=\sum _{i=1}^{n_g}{p}_i,$$(11)

However, it is necessary to study nucleation of each cell in each calculation step. Though, even under variable conditions of nucleation, extended neighborhood of cells remains the same and can be analyzed only once.

4.2. Grain Growth and Carbon Diffusion

The grain growth rate in the model is calculated according to Eq. (7), where boundary mobility (m) and chemical component of driving force (F_c) are calculated as follows:   

$$m=m_{0}exp\left(-\frac{Q}{RT}\right),$$(12)

  

$$F_c={\mu }_{Fe}^{\alpha }-{\mu }_{Fe}^{\gamma },$$(13)

where m₀ = 0.5 [m mol J^–1 s^–1] – coefficient, Q = 147 [kJ mol^–1] – activation energy, ${\mu }_{Fe}^{\alpha },\mathrm{\hspace{0.17em} }{\mu }_{Fe}^{\gamma }$ – chemical potential of iron in ferrite and austenite respectively.

The growth of ferrite is controlled by diffusion and austenite-ferrite interface. Carbon diffusion takes place during the entire process; it can be described as follows:   

$$\frac{\partial C_{\gamma \left(\alpha \right)}}{\partial t}=D_{\gamma \left(\alpha \right)}\left(\frac{{\partial }^2{C}_{\gamma \left(\alpha \right)}}{\partial x^2}+\frac{{\partial }^2{C}_{\gamma \left(\alpha \right)}}{\partial y^2}+\frac{{\partial }^2{C}_{\gamma \left(\alpha \right)}}{\partial z^2}\right).$$(14)

Diffusion in austenite (γ) and ferrite (α) should be considered separately. But diffusion in ferrite do not calculated in the model, because carbon concentration in ferrite essentially less, than in perlite and austenite. Carbon concentration in ferrite is set equal to 0.0217. Diffusion coefficient⁵⁸⁾ is $D_{\gamma }=4.7\times {10}^{-5}exp\left(-1.6C-\frac{3\mathrm{\hspace{0.17em} }700-6\mathrm{\hspace{0.17em} }000C}{RT}\right)$ .

5. Modeling Results

The simulation of microstructure evolution during phase transformations is performed on the same initial microstructure, which is created by cellular automata with 260×260×260 cells and the sizes of 500×500×500 μm³. It contains 100 grains and it is formed by grains growing freely from the randomly distributed nuclei in the space with periodic boundary conditions.

Carbon steel of different carbon contents without alloyed or microalloyed elements was taken for simulation of microstructure evolution during the phase transformation. Steel is considered as dual iron-carbon system. Steel have not been subjected to deformation. Selected steel is cooled from the temperature of 930°C to the temperature of 730°C for 20 seconds. Then, steel is held till full ferrite precipitation (austenite-ferrite transformation). Next, steel is cooled for 100 s with a cooling rate –1 K/s; and finally, it is held till the full austenite-perlite transformation.

Ferrite nucleation was simulated according to section 4.1 with use Eqs. (4), (5), (8) and (9). Growth of ferrite grains and diffusion are described in section 4.2, and then Eqs. (12), (13), (14) are utilized in the model.

5.1. Nucleation Sites

The first results series concerns the questions of differentiation of nucleation location. The question has no clear answer. Various publications indicate preferential nucleation in the grain corners, then on the edges, and finally on the boundaries. Other sources assume the reverse order. They explain that the probability of nucleation at the corners is higher, but such nucleation sites are incomparably fewer than the grain boundaries. As a result, the number of nuclei at the boundaries can be greater (or much greater) than in the corners.

Three extreme nucleation cases have been examined: on the boundaries, on the edges and in the corners. Nucleation conditions are set so that in every case the same number of nuclei arises (approximately 5000). The simulations are carried out for the steel with a carbon content of 0.3 wt%. The results are shown in Fig. 3. In all the colored figures each phase has its own color: red iron oxide corresponds to austenite, canary to ferrite and cobalt to perlite. The boundaries are marked in gray. Actually, gray color in the figures presents state of cells “on the boundary”. If the state of the cell is other, then color represents material state (austenite, ferrite or perlite). Because “boundary” cells lay on both side of the “real” boundary and every cell sizes are bigger than real width of the boundaries, the boundaries in the figures seems much wider than real ones, especially when they cross representative cuboid’s surfaces with low angle. Such wide boundaries are the only results of visualization method and resolution, which depends on the number of cells. In the black-and-white version of the figures, different gradations of grey color correspond to different phases.

Fig. 3.

Effect of nucleation sites on final austenite-ferrite microstructure: nucleation on the boundaries (a), on the edges (b), in the corners (c) of austenite grains. (Online version in color.)

Figure 3(a) presents microstructure after full austenite-ferrite transformation, when nucleation of ferrite grains was on the boundaries of the austenite grains, while Figs. 3(b) and 3(c) show microstructure after nucleation only on the edges and in the corners respectively. Probably, if pictures could not be marked, nobody, including author, could not determine nucleation sites. Insignificant differences in the details allow understanding why the problem of nucleation sites is so complicated, especially when two-dimensional pictures are analyzed and boundaries between ferrite grains are often poorly differentiated. By taking into account the fact that all three variants of the nucleation are presented in the real phase transformation, the question about what kind of nucleation is prevailed, remains open. It requires very accurate and complicated microscopic studies with a three-dimensional microstructure analysis. The results also show that less complicated nucleation algorithm can be used in the model. Consideration of the angles between the boundaries on the edges and in the corners is more accurate, but it is a much more complicated approach, which requires more computational effort and far longer calculation time. That is why, only differentiation on three kinds of nucleation sites without consideration of angles is used in further simulations (algorithm is described in the previous section). The nucleation, when the overall number of nuclei is almost the same on the boundaries, edges and in the corners, is used. The same principle has been also applied for the nucleation of perlite colonies.

5.2. Influence of Carbon Content

Another series of simulations concerns the influence of the carbon content on the microstructure after the phase transformation. Steel with three carbon content of 0.1, 0.4 and 0.73 wt% have been modeled. The results of austenite-ferrite transformation which occurred in the first cooling stage are shown in Fig. 4. The final microstructure after the second cooling stage (after a complete austenite-perlite transformation) is presented in Fig. 5.

Fig. 4.

Austenite-ferrite structure after the first cooling stage for steels with carbon contents of 0.1 (a), 0.4 (b) and 0.73 wt% (c). (Online version in color.)

Fig. 5.

Microstructure after austenite-perlite transformation for the steels with carbon contents of 0.1 (a), 0.4 (b) and 0.73 wt% (c). (Online version in color.)

In the low-carbon steel containing 0.1 wt% C almost 87% of austenite is transformed into ferrite (Fig. 4(a)). While, about 95% of volume remains in the form of austenite in the other steel (0.73 wt% C) closed to eutectoid composition (Fig. 4(c)). In this paper variant when austenite grain boundaries almost do not constitute a barrier for the ferrite grain growth, and therefore the grain growth rate depends little on which austenite grain ferrite grows into, is simulated. Though, in the model, diffusion of carbon through the austenite grain boundaries is somewhat inhibited, and that causes a slight difference in the growth rate of different ferrite grains. Growth is carried out also on the background of a continuous reduction of driving force, which depends on the carbon content in austenite. After the reaching the eutectoid composition, the driving force drops to zero. It is the completion of austenite-ferrite transformation. However, in the model the end of the transformation is indicated by reducing the growth rate to 0.01 of the maximum value at the current simulated temperature. The results of the transformation obtained for the first cooling stage can be considered as satisfactory, corresponding to the real structure not only qualitatively, but also quantitatively.

The results of austenite-perlite transformation in the second cooling stage are presented in Fig. 5. Ferrite formed at the austenite grain boundaries is a natural barrier for the growth of perlite colonies and in low-carbon steel; it reduces the growth of perlite only within a single austenite grain. New perlite colonies are nucleated mainly on the ferrite-austenite sites. As steel contains more carbon, more austenite grain boundaries remain open and then some perlite colonies can grow successfully in several neighboring grains. When steel composition is closed to eutectoid, austenite grain boundaries are very poorly marked in the final ferrite-pearlite structure. For hypereuctoid steel, cementite forms fine mesh on the austenite grain boundaries (not shown in the paper), which is also a barrier for the growth of perlite into several austenite grains, it grows only within a single austenite grain. The formation of cementite is not considered in the model, but for the steel with a carbon content of over 0.8 wt%, pearlite is assumed to grow only within a single austenite grain, where its nucleon appears.

5.3. Carbon Diffusion

In order to demonstrate carbon diffusion during the phase transformation with the same temperature schedule in more detail the cellular space with 600×600×600 cells is used for simulation. Steel with carbon content of 0.1 and 0.6 wt% have been chosen.

Distribution of carbon concentration in two stages of austenite-ferrite transformation is shown in Fig. 6 (carbon content 0.1 wt%) and Fig. 7 (carbon content 0.6 wt%). The Figs. 6(a) and 7(a) present early stage of phase transformation, while Figs. 6(b) and 7(b) the later stage (not full transformation). The lightest areas on the Figs. 6 and 7 are the ferrite grains. At the beginning of the transformation with a low carbon content (Fig. 6(a)), the carbon concentration in the austenite grains is not high, they are slightly different in color from the ferrite grains and only narrow bands around the ferrite grains with much more higher carbon concentration differentiate new growing ferrite grains from the old austenite. Higher carbon concentration is observed around the ferrite grains as well as in the other figures (Figs. 6(b) and 7). Somewhere dark areas are seen in austenite – there are new ferrite grains that do not reach the cross-sections presented by cuboid’s faces. Some of them can be seen as ferrite grains on the later stage of transformation (compare Figs. 7(a) with 7(b)).

Fig. 6.

Carbon concentration at the beginning (a) and the end (b) of austenite-ferrite transformation of steel with carbon content of 0.1 wt%. (Online version in color.)

Fig. 7.

Carbon concentration at the beginning (a) and the end (b) of austenite-ferrite transformation of steel with carbon content of 0.6 wt%. (Online version in color.)

Distribution of carbon concentration along the vertical edge of the cellular space for two stages are presented in Figs. 8 and 9. Figure 8 demonstrate distribution at the beginning of transformation, while Fig. 9 shows it at the end. So, distribution presented in Fig. 8 is correspondent to the carbon concentration along the frontal vertical cuboid’s edge presented in Figs. 6(a), 7(a), while Fig. 9 is relayed to Figs. 6(b) and 7(b), but instead of color they show value of carbon concentration. In Fig. 8 it is seen three main levels of carbon concentration: low (0.0218) for ferrite, and two initial levels for austenite (0.1 and 0.6). Near the ferrite the high step of carbon concentration can be observed; concentration can reach value of 0.77. Somewhere: near the cell number 360 for low carbon steel and near the cell numbers 100, 200, 420, 290 and 570 for the higher carbon steel increase of carbon concentration can be seen. There is impact of growing ferrite grains, which do not reach the vertical line represented in the figures.

Fig. 8.

Carbon concentration at the beginning of austenite-ferrite transformation.

Fig. 9.

Carbon concentration at the end of austenite-ferrite transformation.

In Fig. 9 for the low carbon steel, almost the final state is established, when only two levels of carbon concentration are presented: 0.0218 for ferrite and 0.77 for austenite. For steel with the higher carbon content, the concentration in austenite is higher than the initial value (0.6), but it does not reach the final equilibrium value (0.77).

The highest carbon concentration is observed in front of growing ferrite grains. Gradient of carbon concentration in austenite near the boundaries of growing ferrite grain in steel with low carbon content is much higher at the beginning, than at the end or in the steel with the higher carbon content. It corresponds to the theory and observations.

5.4. Comparison 3D Model with 2D

Average ferrite grain size and volume fraction for three different carbon contents are presented in Fig. 10. Fourier number Fo was used as a time axis: Fo = DΔt/R², where D – diffusion coefficient, Δt – time, R – average radius of austenite grain. Two differences can be seen. The first one is final ferrite grain size. Final volume of the grains is the same (Fig. 10(b)), but sizes are different and depend on carbon content. When steel with higher content is simulated, final grains are finer. Volume of fine ferrite grains is close to πD_α³/6 and πD_α²/4 for 3D and 2D respectively; and grain occupies π/6 and π/4 part of cube or square of D_α sizes. Then the one 3D grain of the same diameter gives less volume and volume fraction, than 2D grain. Therefore 3D grain should be bigger, than 2D grain, for the same volume fraction. For the steel with lower carbon content, the small ferrite grains can be replaced by the untransformed austenite in the explanation; and it give bigger size for 2D ferrite grains.

Fig. 10.

Changes of ferrite grain size (a) and volume fraction (b) during the transformation.

The other difference is observed in kinetics of transformation. Because transformation is controlled by diffusion and excess carbon from the boundary spreads faster in three dimensions, than in two; 3D transformation is the fastest at the beginning. It also can be seen in Fig. 10(b).

6. Summary

The developed FCA based model is used for a simulation of phase transformation. FCA are implemented into the hierarchical model for simulation, study and prediction of the microstructure evolution in technological processes. The phase transformation is considered as the nucleation and the growth of grains of other phases. In the paper, the main attention is paid to the development of nucleation algorithm and study of the effect of nucleation sites on final microstructure. It was shown that a problem of nucleation sites is very difficult to solve by experimental microscopic methods and simulations, and it is based mainly on theoretical analysis. Simulations of phase transformation are carried out with the low cooling rate and the holding for a relatively long time at constant temperature, which allows firstly to complete a full austenite-ferrite transformation, and then to obtain a full austenite-perlite transformation. A low cooling rate leads to a globular shape of growing ferrite grains.

Carbon diffusion as a main mechanism controlling the transformation was also studied. The simulation results confirm the great possibilities and effectiveness of the FCA model for modeling of the microstructure evolution during the phase transformation.

Results presented in the paper concern preliminary simulations of phase transformation, which mainly present ability of developed tools to model complex phenomena taking place during the transformation. To validate the models of nucleation, grain growth, controlled by diffusion, and influence of process conditions (initial microstructure, cooling rate, carbon content, deformation and so on), experimental studies are planned: dilatometric, microscopic and lately plastometric tests.

Acknowledgments

The author is grateful to the Polish Ministry of Science and Higher Education for the financial support (project no. N508 3812 33).

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