Simulating the kinetics of recrystallization in Aluminum alloys

This contribution presents a brief overview of the simulation of recrystallization kinetics in metals. The study focuses on the annealing phenomena in a single-phase 1050 aluminum alloy of technical purity. To reveal the true nature of recrystallization, the kinetics is discussed by the well-establishes Johnson–Mehl–Avrami–Kolmogorov (JMAK) approach, while the constants of this model are related to the energy stored during deformation, nucleation rate, velocity of grain boundary, and grain boundary energy. The listed physical quantities are derived from different models, while the performance of the combined approach was tested for the cases where the diversity of driving forces for recrystallization was ensured by different straining levels. The softening of the material during annealing was evaluated by the microhardness. It was shown that the kinetics of recrystallization is strongly influenced by the stored energy and the process can be simulated by employing the JMAK equation.


Introduction
In metal processing, the materials are subjected to complex thermomechanical processing (TMP), which typically involves deformation and annealing.The forming of materials such as forging, extrusion, or rolling tends to increase the number of linear defects called dislocations, which in turn lead to hardening.In order to recover the physical properties after deformation, the metals are subjected to annealing.The heat treatment can be performed at various temperatures, while the degree of recovery depends on annealing time.During the low-temperature annealing (~ 1/4 to 1/3 of melting point Tm) the rearrangement of dislocations is observed without a significant drop in dislocation density, whereas at the temperatures of ~ 1/2Tm, the process of recrystallization (RX) takes place.Recrystallization starts with nucleation after a certain incubation period, and the new low-energy domains/nuclei consume the high-stored energy matrix.The resulting microstructure differs significantly from the deformed counterpart and in highly strained materials the annealing induces grain refinement due to the high population of nuclei.The process of transformation from a deformed state to recrystallized one is generally described by the well-known Johnson-Mehl-Avrami-Kolmogorov (JMAK) theory [1][2][3][4][5][6]: where   is the fraction of recrystallized grains, B is a constant, t is the time of annealing, and n is a so-called Avrami exponent.
The above equation was derived almost a century ago and the model parameters were defined for numerous metallic systems [1].As it turned out, the constants B and n are complex functions of (i) microstructural features, which tend to evolve during deformation, (ii) thermomechanical processing parameters, and (iii) chemical composition [1][2][3][4][5][6].The TMP variables such as the degree of straining, deformation temperature, annealing temperature, and holding time strongly affect both the kinetics of recrystallization and the final microstructure and also influence the values of the JMAK constants.For instance, an increase in the deformation temperature induces variations in the nucleation and subsequent growth rates of the nuclei, thus, accounting for changes in the JMAK model parameters.In equation 1, the exponent n is related to the mechanism of nucleation, while the constant B is a function of nucleation rate N (depends on chemical composition) and annealing temperature.
In general, the exact relationship between the TMP parameters and the JMAK tuning values is complex and depends on the specific material and processing conditions.In view of the fact that many properties are microstructure controlled, it is important to employ numerical models, which can help identify the correlation between the experimental evidence and simulations and provide insights into the challenging issue related to the optimization of processing parameters for desired microstructural state and properties.
Even though a number of investigations dealing with recrystallization have been performed [2][3][4][5][6][7][8][9], however, the comprehensive mathematical description, allowing the accurate simulation of the process is still missing.In view of this, this contribution presents a concise summary of the algorithm, enabling the simulation of recrystallization kinetics.In the computational procedure described, the JMAK constants are expressed by physical quantities, which can be calculated by employing physically based models.The major goal of this contribution was to develop a technique that enables the simulation of RX kinetics, with a minimum number of fitting parameters.Since the described approach is of generic nature, it can be implemented not only for Al alloys but also for any arbitrary metallic system.

Materials and experimental procedures
The investigated 1050 alloy is a commercially available aluminum of technical purity from 1xxx series (Al content is min.99.5 wt%).This alloy typically contains traces of iron (~0.3 wt%) and Si (~0.2 wt%), while the concentration of other alloying elements such as Cu, Mn, Mg, V, and Zn is negligibly small (~0.05 wt%).The high purity ensures a single-phase state with a minimum amount of second phases.It was important to select a single-phase material with the aim to provide relatively homogeneous strain distribution during deformation since the hard inclusions act as stress risers and lead to so-called particle-affected deformation zones.
To ensure a diversity of driving forces for recrystallization, the as-received 1050 Al sheets were annealed at 550°C for 10 minutes to guarantee a fully recrystallized state and afterward, the heat-treated samples were cold rolled with different thickness reductions: 36% (sample A), 46% (sample B) and 54% (sample C).The single-pass cold rolling trials were performed by a laboratory rolling mill with a roll diameter of 150 mm.The deformed samples were annealed at 450°C for different times, ranging between 1 and 25 minutes.This allowed us to investigate the annealing phenomena at various stages of recrystallization.
In order to examine the kinetics of softening process, the hardness values were measured at different stages of recrystallization.The indentation was performed by using Zwick/Roell ® ZHVμ-type Vickers microhardness tester.During the microhardness measurements, the diamond-shaped indents were made on the transverse direction planes of investigated samples.To avoid the consequences of the Indentation Size Effect (ISE) [10,11], the measurements were performed with high loads (falling into the saturation zone of the ISE curve).In the current study, a load of 19.62 N was employed, and in this way, the hardness impact from the maximum number of grains was recorded.
Prior to indentation measurements, the investigated samples were prepared according to the standard metallographic procedure, which involves mechanical grinding, polishing as well as electrolytic polishing.The mechanical polishing was finished with two DiaDuo Struers®-type suspensions which contain 3 and 1 m diamond particles, respectively.After achieving a mirror-type surface quality, the investigated samples were cleaned with liquid soap, rinsed with, and dried using a blow drier.The electrolytic polishing was conducted with A2 Struers® electrolyte for 45-60 seconds at voltages ranging from 20 to 30 V. The electrolyte was cooled to temperatures ranging between -5 and 0°C.

Results and Discussion
The JMAK model parameters can be related to thermomechanical processing variables by investigating the deformed and recrystallized states of materials.

Defining model parameters from the deformed state
The process of deformation is accompanied by the generation of dislocations.When the material is subjected to straining, the dislocations tend to move and interact with each other, which causes energy to be stored within the material.This elastic energy is related to the dislocation density as [1]: where α is a strain-dependent geometric constant, G is the shear modulus (G=26.5 GPa), and  is the dislocation density, and b is a Burgers vector (b=0.2863nm).
The geometric constant α is usually assumed to be independent of strain (α~0.5),however, this quantity depends on the mean free path of dislocations, which is inversely proportional to the dislocation density, and can be computed by the following relation [1,11]: ( where  is a Poisson ratio (=0.35).To calculate both α and ED, the dislocation density should be assessed.Generally, the  is estimated by means of transition electron microscopy, however, in many instances this complex investigation is replaced by numerical modeling, which is capable of reproducing the experimentally measured counterparts with satisfactory accuracy.The approach developed by Csanádi et al. [12], enables the computation of  as a function of applied strain e: In the above expression, the dislocation density of deformation-free material 0 is assumed to be 10 10 m -2 , whereas the constants C1 and C2 for aluminum are: C1 = 2.33 ×10 14 m -2 and C2 =1.15 [11].Figure 1 presents the dependency of dislocation density on the straining level calculated by equation 4. It is obvious that even a low amount of deformation tends to induce a drastic increase in dislocation density.The generation of dislocations slows down as the deformation progresses due to dislocation interactions and their trapping.The same evolutionary pattern is characteristic for the stored energy since ED ~ .

Kinetics of recrystallization: simulation and experimental evidence
To simulate the kinetics of RX process, it is of key importance to analyze the technological and physical parameters which are related to the JMAK constants.The JMAK equation is time dependent expression, while the temperature effect is accounted for via constant B, which is mathematically expressed as [1]: where N is the nucleation rate,   is the growth rate of recrystallized domains (dislocation-free areas), and f is the shape factor (f = 4 3 ).The accumulation of deformation energy has a direct effect on recrystallization since the growth rate of the high-angle grain boundary (between the recrystallized and deformed grains) is a function of stored energy.The process of growth is also related to the mobility of the high-angle grain boundaries and therefore, the growth rate is expressed as follows [1,13]: where M0 is a material constant (M0=3.1×10 - m 4 /Js), QG is the activation energy required for the migration of high-angle grain boundaries (QG = 8.01×10 -20 J), T is annealing temperature and kB is Boltzmann constant [13,14].
Generally, the nucleation rate N refers to the frequency at which nuclei tend to form from the deformed matrix.The value of N is affected by several factors, including the straining level and temperature.At low strains, the N is generally low since there is a limited amount of linear defects in the material (compared to highly strained counterparts) to act as nucleation sites, and therefore the process of grain growth is dominating.As the strain level increases, the dislocation density also tends to rise (see figure 1) and therefore the probability of nucleation will be higher.In the metal matrix deformed with strain e, the number of nuclei generated per unit volume within a certain time frame can be expressed as [14]: where N0 is the temperature-independent preexponential factor (N0 =2.87×10 15 m -3 s -1 ) and g surface energy of the grain boundaries.
The value of g can be estimated by the Read and Shockley equation [1,15]: with where   is the radius of dislocation core (   = 3 4 ) and θ is the misorientation angle (θ = 15°) [15,16].
The estimated value of g according to the above equations is 0.192 Jm -2 and is comparable to one claimed in literature sources (up to 0.3 Jm -2 ) [13,14].
Combining equations 5-10, one can calculate the dependence of JMAK model parameter B on TMP parameters such as annealing temperature or deformation level.In industrial practice, the thickness reduction may vary in a wide range; typically, from 20% to 80%. Figure 2 shows the variation of B and nucleation rate N with strain for the annealing temperature of 450°C.It is evident that both values are strongly impacted by the degree of rolling reduction.To simulate the kinetics of RX by equation 1, it is necessary to define the Avrami exponent n.The JMAK equation was originally derived by assuming random nucleation, i.e. the nuclei tend to develop randomly in the deformed matrix.In this case, the value of n is equal to 4 and this event is known as site-saturated nucleation.The kinetics of RX with n=4 was observed in fine-grained materials subjected to low straining levels [1], when the deformed grains produced only one nucleus (or even 0).A vast variety of experimental evidence dealing with the kinetics of RX claims that n is far below 4 [1,8,17,18].The discrepancy can be attributed to the heterogeneous nature of deformation in a textured polycrystalline matrix, the development of microstructural inhomogeneities, the occurrence of recovery, and the fact that the driving force is not constant during recrystallization.
There are numerous methods enabling the examination of softening kinetics experimentally, such as optical microscopy, electron backscattering diffraction, or indentation techniques.In the present study, the evolution of recrystallization is traced by means of Vickers microindentation.Let's assume that the measured hardness Hi is composed of deformation component Hd (with the volume fraction Xd) and recrystallization counterpart Hrx (with the volume fraction Xrx): =     +     =   (1 −   ) +     (11) By rearranging equation 11, one can obtain the fraction of recrystallized grains Xrx: Figure 3 shows the experimentally observed and simulated RX kinetics with the constant B, shown in figure 2 and Avrami exponent of 2.5.The simulated kinetics by equations 1-10 can capture the effect of straining level on the evolution of recrystallization.The correlation coefficient between the experimentally observed and simulated data is R 2 =0.98.It is also obvious that the simulated curve shows some discrepancy with respect to the experimental evidence proving that nature of recrystallization is complex, and needs to be further investigated.

Conclusions
Examination of recrystallization kinetics after different straining levels shows that the process can be reasonably described by the Johnson-Mehl-Avrami-Kolmogorov theory.The model parameters used for the simulation were derived from different numerical approaches.
It is shown both experimentally and by employing numerical approaches that the RX kinetics is strongly influenced by the degree of straining.
The discrepancy between the modeled and simulated counterparts can be attributed to the heterogeneous nature of deformation in a textured polycrystalline matrix, the development of microstructural inhomogeneities, the occurrence of recovery during annealing, and the fact that the driving force is not constant during recrystallization.Although all mentioned phenomena were neglected during the computations, the simulated curves still reproduced the experimentally observed kinetics with reasonable accuracy (R 2 =0.98).

Figure 1 .
Figure 1.Calculated dependence of dislocation density on the straining degree.

Figure 2 .
Figure 2. Calculated dependence of JMAK constant B and nucleation rate N on the degree of rolling reduction for 1050 Al alloy annealed at 450°C.