Introduction

Grain growth after primary recrystallization is described as taking place in two different forms, either in form of continuous grain growth (“normal”) or in form of discontinuous grain growth (“secondary recrystallization”). In the development of this area two stages can be distinguished. In the first stage simple and mostly qualitative interpretations of grain growth have been given. For continuous grain growth, Beck [2] predicted an increase of the average grain diameter with time as t^1/2which was practically never found. The second stage of the development of this area is characterized by more sophisticated approaches. Hillert [3] transferred the statistical treatment of Ostwald ripening of precipitates to grain growth according to the Lifshitz-Slyozov-Wagner theory [4-5]. Moreover, Hunderi and Ryum [6] introduced a deterministic model considering individual boundaries and describing the change of size of the individual grains by an extremely large set of differential equation (one for each grains) which they solved numerically. Finally, Abbruzzese [1] developed further the Hillert model by calculating a critical radius that was only postulated by Hillert [3]. The main novelty of Abbruzzese study was to use discrete grain size classes which reduce significantly the numbers of differential equations (one for each class) and thus to the possibility to calculate numerically the evolution of the grain size distribution. In recent years the Monte-Carlo simulation was widely used to simulate grain growth including also the case of Zener drag [7]. Hesselbarth and Gobel used, with success, a method named cellular automata in simulation of the theory of Johnson-Mehl-Avrami-Kolmogorov and other successful mesoscale simulations for microstructural evolution including front tracking model [8], vertex model [9] and phase field model [10] have been developed. While analytical models, such as Abbruzzese and Lücke, predict all the characteristics of microstructural evolution (i.e., grain size and grain size distribution), the goal of mesoscale computational simulations is rather different: to generate snapshots of the evolving microstructure with time. Using the computational version of metallography, both local and ensemble properties of the microstructure may be determined from these snapshots.

Description of the model

A mathematical model able to simulate simultaneously recrystallization and grain growth phenomena is described in this paper. The driving force for primary recrystallization in metals is mainly related to the reduction of the deformation energy (dislocations) introduced by cold working. Heat treatment activates the movement of dislocations and sub-grain boundaries allowing the release of the deformation energy and thus restoring a “dislocation free” microstructure. Under further heat treatment, grain growth activated by boundary energy reduction is the dominant process [7]. In this approach, recrystallization nuclei are considered pre-existing and homogeneously distributed in the deformed microstructure.

As far as concerns grain growth, the statistical model, originally developed by Abbruzzese and Lucke [1], is based on the assumptions of:

Super-position of average grain curvatures in individual grain boundaries. A grain v is characterized by a volume V, is assumed to growth at the expense of a neighbouring  grain µ with a rate:

Here the “radius” R_v of a grain v is defined according Vv= 4pRv³ is the area of contact between the two grains v and µ, and m, γ and M=2m γ represent respectively, the mobility, the tension and the diffusivity of the grain boundary vµ. By taking into account all Nv neighbours of this grain one obtain for all its total growth rate:

With v=1, 2….N_G and N_G being the total number of grains, this expression represent a system on N_G differential equation for the unknown R_v. However, because of the large numbers of grains N_G and thus a large number of equation is necessary to obtain a significant simulation, this leads to a great computational difficulties. Therefore, with the second and third simplifying assumptions will be easy to overcome these difficulties:

Homogeneous surroundings of the grains. As a first approximation is assumed that for each grain v the individual neighbourhood of N_v individual grain can be replaced by a surrounding obtained by averaging over a neighbourhood of all grains of the same radius R_v. Since then all grains of the same radius would have the same surrounding, also their growth rate would be equal. This means that then all grains could be collected in classes characterized by their radius and that the behaviour of only different classes has to be considered, instead of single grains. In the following these classes will be denoted by the indices i, j…N_s being the total number of classes. From the mathematical point of view the simplification consists in replacing in Equation (3) the individual contact area svu by averaged area aij = Aij/nj,nj is the total number of grains in class j and A_ij is the total area of contact between the two classes i and j. Then, it follow that:

A random array of the grains namely the probability of contact among the grains is only depending on their relative surface in the system. In this case the area of a grain of the class i is divided between the neighbouring grains of the class j in proportion to the individual surface area:

The integration of all the above assumptions in the model leads to the following final form of the grain growth rate equation:

Where M = 2my is again the boundary diffusivity and in our case study m was evaluated according to the Stokes-Einstein relationship [11]:

Where D is the diffusion coefficient, K_B is the Boltzmann constant, ΔE is the activation energy of the process and T is the annealing temperature. D was chosen proportional to the diffusion coefficient of Fe in Fe-γ.

According to Zener [7], particles cause inhibition of grain boundary motion which can be considered as a retarding forces acting homogeneously along the moving boundary and has a similar behaviour of a frictional forces. In order to apply this description to grain growth phenomena has to be taken into account the magnitude of this force P_Z. In fact, for a small external driving force P no motion of the boundary takes place. Only if external force surpasses a maximum force I_Z0 the particles can exert, the boundary moves, but with a net driving force ΔP=P-I_Z0. This lead to three ranges for this net force acting on boundary [12]:

For the maximum Zener force I_Z0 the will be used the usual expression where f_p is the volume fraction is the volume fraction _and r_p is the mean radius of the particles; β is a proportional constant in the range of 0.75-1

With equation (7) and (8), one obtains the net force ΔP_ij in the three ranges:

To describe the recrystallization process integrated with the grain growth, it is necessary to propose an extended growth equation that allows to contemporarily and continuously analyse the evolution of free nuclei in the matrix passing through partially impinged grains up to full contact. Was introduce an “influence mean radius” that allow to evaluate the fraction of surface in contact between different grain [13].

The final equation for recrystallization and grain growth can therefore be written as:

Where G is the shear modulus of the material, b is the Burger vector, ρ is the dislocation density, Δρ=ρ_d-ρ_r is the difference between the dislocation densities in the deformed and in the recrystallized material. The dislocation density ρ was considered proportional to the cold reduction rate and the initial numbers of nuclei N is a free parameter of the model that change in relations to reduction rate.  The criterion for identifying the critical class is obtained by defining an average influence volume, and consequently an influence radius R_m, calculated as follows:

N_T  is the numbers of grain per cm³ , F_V is the recrystallized volume fraction and v_i is the volume of the grain of the class and n_i is the number of grains of volume v_i . R_m varies from (4pN_T) to zero when all the grain are in contact. Then R_m parameter define an index i^* that discriminates the class over which all the grains are in contact. [13]

Results and discussion

Annealing temperature effect

One of the free input parameters of the model is the annealing temperature deeply influencing the mobility parameters m and consequently affecting the grain size and the recrystallized volume fraction. Four temperatures, ranging from 700°C to 1100°C, have been investigated and reduction rate ε, dislocation density Δρ and the number of nuclei N was maintained constant. The effect of annealing temperature on grain size is reported in Figure 1. Results show that the mean radius size decreases with decreasing temperature because due to a decrease of the mobility parameters from 3.16e-11 erg/cm² at 1100°C to 3.23e-14 erg/cm² at 700°C, according to Equation 6. Results also show that a minimum temperature of 700°C is required in order to activate grain growth.

Figure 1: Mean radius over time for four different annealing temperature.   Click here to view figure

Also, the recrystallized volume fraction (Figure 2), reach for the two temperatures 1100°C and 1000°C the unitary value respectively in 13 seconds and 49 second and for the other two temperatures this value is never reached.

Figure 2: Recrystallized volume fraction over time for four different temperatures.   Click here to view figure

Reduction rate effect

For industrial purposes and for the statistical model, the reduction rate is one of the most important parameters and in Table 1 the different values used for the computer simulation are shown. The effect of reduction has been exploited by varying the reduction rate ε (at constant temperature equal to 1110°C) jointly to the dislocation density, according to Table 1.

Table 1: Dislocation density and initial number of nuclei used as input parameters in the simulations.

Figure 3: Mean radius over time for three different reduction rate.   Click here to view figure

The simulation results confirm that if cold reduction is varied from 40% to 80%, mean radius increase about 4%. This could be explained considering that higher values of deformations introduce in the structure more deformation micro-cells (nuclei N) that could increase less their size due to the excessive proximity of neighboring nuclei.

Zener drag effect

Three different value of Zener parameter has been tested maintaining temperature (T=1100°C), reduction rate (ε=80 %) and dislocation density (Δρ=1x10^11 cm^-2) fixed. The comparison between the mean radius curve (over time) obtained by grain growth simulation without Zener effect and the other curves are shown in Figure 4. At the highest Zener parameter Iz=1000 a reduction of the maximum mean radius is clearly visible and is equal to 41.39 % and in Figure 5 the maximum mean radius values in relation with Zener parameter are reported.

Figure 4: Mean radius over time for three different values of Zener parameter.   Click here to view fiugre

Figure 5: Maximum mean radius in relation with different Zener parameters.   Click here to view figure

Conclusions

Results from a recrystallization and grain growth model based on statistical assumption have been here discussed. In particular, the effect of the annealing temperature, dislocation density and Zener parameter have been analyzed. Results show that:

• mean radius size decreases with decreasing temperature because due to a decrease of the mobility parameters;
• mean radius increase about 4% with cold reduction variation in the range 40% to 80%;
• the model is able to take into account Zener effect.

References

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