# A Multilevel Physically Based Model of Recrystallization: Analysis of the Influence of Subgrain Coalescence at Grain Boundaries on the Formation of Recrystallization Nuclei in Metals

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mechanisms of Subgrain Structure Evolution under Thermomechanical Effects

## 3. Materials and Methods

_{sb}at the grain facet, the energy E

_{cl}, decreasing locally during coalescence, is calculated as:

_{0}is the initial representative volume of the polycrystal, ${\mathrm{V}}_{r}$ is the recrystallized material volume, ${N}_{r}$ is the number of recrystallized grains, and ${\mathrm{v}}_{r}^{\left(i\right)}=\frac{4}{3}\mathsf{\pi}{r}^{\left(i\right)}{}^{3}$ is the volume of the i-th sphere of the recrystallized grain. Note that the initial volume of the recrystallized grain coincides with that of the recrystallization nucleus (subgrain) for which criterion (1) has been fulfilled.

## 4. Results and Discussion

^{−3}s

^{−1}[88], and (2) at temperature 300 K and deformation velocity 2·10

^{−3}s

^{−1}[89]. For the high-temperature experiment [89], the hardening model parameters were determined prior to the active stage of dynamic recrystallization, i.e., at about 20% deformation. Computational experiments performed at deformation values obtained before the onset of active dynamic recrystallization, i.e., at 10–15% deformation, were also considered.

^{−3}s

^{−1}and temperature $T$ = 550 K) was carried out. In the reference configuration, the special boundary ${\mathsf{\Sigma}}_{3}$ corresponds to the mutual misorientation of the grains rotated by an angle of 60° with respect to the general direction [111]. According to (9), the value of the energy of this grain boundary was 0.225 J/m

^{2}. As the grains deform, they undergo rotations determined by the applied rotation model [83], and the initial special boundary becomes the high-angle grain boundary with a small number of coinciding sites. Figure 7a illustrates the evolution of the grain boundary energy for the initial special boundary 60° [111]; at 5% deformation, the special boundary ceases to be such and its energy increases to that of the high-angle boundary. Figure 7b,c presents the dependencies of the linear mean size of subgrains <d

_{sb}> of the considered grain with high-angle (Figure 7b) and special ${\mathsf{\Sigma}}_{3}$ (Figure 7c) boundaries. In accordance with the identified parameters [34], the high-angle grain boundary energy ${e}_{gb}$ was 0.337 J/m

^{2}. By analyzing the obtained results, the “boundary” (adjacent to high-angle grain boundary) subgrains and the coalescence-induced boundary subgrain clusters were assembled into one group, and the “inside” (non-adjacent to high-angle grain boundary) subgrains into another group (Figure 5). The results given in Figure 7b confirm the previous assumption that coalescence at arbitrary high-angle boundaries develops more intensively compared to the remaining grain volume or at special boundaries (Figure 7c). Thus, the clusters of subgrains with increased dimensions are formed at arbitrary grain boundaries. These clusters serve as energetically favorable nuclei (sites) for further recrystallization. No such effect was observed at a special boundary (Figure 7c). At the end of deformation, the mean subgrain size was 0.375 µm at the incident boundary and 0.270 µm at the special boundary. The evolution of the resulting subgrain cluster at the considered arbitrary grain boundary is demonstrated in Figure 7d for different instants of deformation. In general, the random orientation of the bicrystal with an incident high-angle boundary does not change the nature of the dependencies given in Figure 7; the same is true for the special boundary.

^{−5}s

^{−1}and $T$ = 550 K. Figure 8b presents the same data for the special boundary ${\mathsf{\Sigma}}_{3}$. Similar histograms are given in Figure 8c,d for the case when coalescence is neglected; the high-angle boundary type has no effect on the results. The analysis of the results revealed that almost the same large subgrain clusters as those inside the grain are formed at the special boundary; that is why they will not be considered below.

^{−5}s

^{−1}to 10

^{−3}s

^{−1}and temperatures $T$ from 550 to 775 K. Figure 9a shows how the mean subgrain size changes in the specified ranges of temperatures and deformation velocities. The points corresponding to the onset of recrystallization and the 5% volume fraction of recrystallized material are denoted by symbols “${r}_{0}$” and “${r}_{5}$”, respectively. In this case, the growth of the mean subgrain size significantly depends on the coalescence process, which is actively implemented at elevated temperatures and low velocities of deformation. It is important now to pay attention to the non-monotonic behavior of the plot displaying the dependence of $<{d}_{sb}>$ on the deformation intensity. This behavior is explained by the fact that large subgrains are assigned to the category of individual recrystallized grains, provided that criterion (1) is fulfilled. The evolution of the average size of recrystallized grains $<{d}_{gr}>$ is shown in Figure 9b. The nature of the nonmonotonic curve of the function $<{d}_{gr}>$ (Figure 9b) is caused by two processes: the transition from subgrains to individual recrystallized grains, and the normal growth of recrystallized grains. Note that the recrystallization criterion (1) is fulfilled first for coarse subgrains. For these subgrains, the effect of the energy of the grain boundary (second term in (1)), at which the recrystallization process slows down, is less pronounced. Thus, a sharp increase in the size of recrystallized grains $<{d}_{gr}>$ is seen on the graph at the initial instant of recrystallization (Figure 9b). At low deformation velocities of 10

^{−5}s

^{−1}, the growth of recrystallized grains proceeds more intensively than the subgrain transition as a result of fulfilling the recrystallization criterion for small subgrains; the average grain size increases. At 10

^{−5}s

^{−1}, the rate of transition from subgrains to recrystallized grains exceeds the normal grain growth in a certain deformation segment. Based on the results obtained, it can be concluded that the low deformation velocities and elevated temperatures promote both an increase in the size of subgrains at coalescence and the formation of new recrystallized grains.

^{−5}to 10

^{−3}s

^{−1}and at $T$ from 550 to 775 K is given in Figure 10. The value of recrystallized material ${X}_{r}$ is determined by relations (10)–(12). Figure 10a presents the results of modeling obtained with the consideration of coalescence, and Figure 10b shows these results ignoring coalescence. It can be seen that coalescence induces recrystallization at lower values of deformation and is responsible for the intensive growth of new grains.

^{−5}s

^{−1}, T = 700 K) changes in the bicrystal grains under study. The evolution of the volume fraction of recrystallized material is demonstrated in Figure 11b. Since the initial stored energy depends on the density of subgrain boundary dislocations, then, for equal subgrain misorientations, the difference ${\widehat{e}}_{dst}$ is practically equal to zero. Additionally, vice versa, in the case of varying initial defect structure, this value was different from zero; its subsequent decrease, shown in Figure 11a, is associated with the coalescence-induced stored energy release. The increase in stored energy, visible on all graphs in Figure 11, is associated with the accumulation of defects inside grains. Despite the fact that coalescence is a process in competition with recrystallization regarding the stored energy, an increase in the level of stored energy eventually leads to the fulfillment of the recrystallization criterion (1). The onset of recrystallization and its subsequent evolution is illustrated in Figure 11b. Recrystallization also causes a decrease in the stored energy, which in turn reduces the rate of accumulation of ${\widehat{e}}_{dst}$ (Figure 11a). It can be seen that the implementation of coalescence with equal intensity—“same coalescence”—(the scenario from Figure 3a) slows down recrystallization and increases the critical deformation compared to the coalescence occurred in an inhomogeneous fashion—“various coalescence”—in adjacent grains (scenario from Figure 3b). In both cases, coalescence promotes the earlier onset of recrystallization compared to the case when the coalescence event is ignored. For comparison, the results corresponding to the “no coalescence” scenario are shown in Figure 11a.

## 5. Conclusions

^{−5}s

^{−1}and 700 K, 10

^{−5}s

^{−1}and 775 K, 10

^{−3}s

^{−1}and 775 K, and 10

^{−3}s

^{−1}and 700 K), respectively. To evaluate the above-described effect of the storage energy release during coalescence, the recrystallization criterion was modified with regard to the SIBM mechanism (Equation (8)). This provides a possibility for evaluating a decrease in the coalescence-induced stored energy. In addition to the energy aspect, coalescence leads to the growth of subgrain sizes. It is shown that coarse subgrain clusters, which further become recrystallization nuclei, are formed at high-angle grain boundaries (Figure 9a). High coalescence intensities at favorable conditions (low deformation rates and high temperatures) contribute to an increase in the average subgrain size from 0.25 µm to about 1.7 µm (Figure 9a). This leads to an increase in the growth rate of recrystallized grains (Figure 9b), which reaches a maximum value of 0.0033 µm/s in the case of deformation with a rate of 10

^{−5}s

^{−1}and a temperature of 775 K, and minimum 0.00013 µm/s for 10

^{−3}s

^{−1}, 700 K. Unlike arbitrary boundaries with energy of 0.337 J/m

^{2}, these effects do not appear at special boundaries with reduced energy 0.225 J/m

^{2}, and thus there is no positive effect of special boundaries on coalescence and the formation of recrystallization nuclei. The results of the multilevel modeling of inelastic deformation in the example of a copper bicrystal demonstrate the capabilities of the developed model for describing the material substructure evolution and the influence of coalescence on the recrystallization process. This model is a component of the multilevel one for describing behaviors of representative volume elements of polycrystalline materials.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Scheme of the formation of a recrystallization nucleus according to the SIBM mechanism (the drawing is based on the scheme given in [46]).

**Figure 2.**Scheme of subgrain rotation at coalescence (based on the scheme from [41]).

**Figure 3.**Scheme of the formation of a recrystallization nuclei at high-angle boundaries [52]: (

**a**) intensive coalescence in adjacent grains and (

**b**) in one of the grains.

**Figure 4.**Scheme: scale levels and relation between the multilevel model structural elements (thin lines—subgrain boundaries, thick lines—grain boundaries) [27].

**Figure 5.**Scheme of the formation of a representative volume of subgrains for the grain boundary facet under consideration.

**Figure 6.**(

**a**) Face-centered crystal lattices misoriented towards the [111] direction by the 60° angle; (

**b**) superlattice which corresponds to this special orientation.

**Figure 7.**(

**a**) Evolution of the grain boundary energy, which corresponds, in the reference configuration, to the special boundary at 60° [111]; (

**b**) evolution of the linear mean size of subgrains $<{d}_{sb}>$ placed in different grain parts with respect to the high-angle boundary for high-angle and (

**c**) special boundaries; (

**d**) evolution of the subgrain cluster during coalescence at high-angle boundary, obtained in the numerical quasi-uniaxial deformation experiment ($\dot{\mathsf{\epsilon}}$ = 10

^{−3}s

^{−1}, $T$ = 550 K).

**Figure 8.**Linear subgrain size distribution histograms plotted based on the results obtained in the framework of the multilevel model with consideration of coalescence at (

**a**) high-angle and (

**b**) special boundaries and without consideration of coalescence at (

**c**) high-angle and (

**d**) special boundaries during the quasi-uniaxial deformation test ($\dot{\mathsf{\epsilon}}$ = 10

^{−5}s

^{−1}and $T$ = 550 K).

**Figure 9.**(

**a**) Evolution of the mean subgrain size $<{d}_{sb}>$ and (

**b**) recrystallized grains $<{d}_{gr}>$ in the uniaxial deformation experiment at different deformation velocities and temperatures.

**Figure 10.**Dependence of the volume fraction of recrystallized material ${X}_{r}$ on the strain intensity obtained in the experiment on uniaxial deformation at different deformation velocities and temperatures: (

**a**) coalescence is considered and (

**b**) coalescence is ignored.

**Figure 11.**Evolution of the difference in specific stored energy ${\widehat{e}}_{dst}$ associated with (

**a**) coalescence and (

**b**) the volume fraction of the recrystallized material ${X}_{r}$ on the deformation intensity in the uniaxial deformation experiment.

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**MDPI and ACS Style**

Trusov, P.; Kondratev, N.; Baldin, M.; Bezverkhy, D. A Multilevel Physically Based Model of Recrystallization: Analysis of the Influence of Subgrain Coalescence at Grain Boundaries on the Formation of Recrystallization Nuclei in Metals. *Materials* **2023**, *16*, 2810.
https://doi.org/10.3390/ma16072810

**AMA Style**

Trusov P, Kondratev N, Baldin M, Bezverkhy D. A Multilevel Physically Based Model of Recrystallization: Analysis of the Influence of Subgrain Coalescence at Grain Boundaries on the Formation of Recrystallization Nuclei in Metals. *Materials*. 2023; 16(7):2810.
https://doi.org/10.3390/ma16072810

**Chicago/Turabian Style**

Trusov, Peter, Nikita Kondratev, Matvej Baldin, and Dmitry Bezverkhy. 2023. "A Multilevel Physically Based Model of Recrystallization: Analysis of the Influence of Subgrain Coalescence at Grain Boundaries on the Formation of Recrystallization Nuclei in Metals" *Materials* 16, no. 7: 2810.
https://doi.org/10.3390/ma16072810