1 Introduction

Reducing grain size to nanoscale in polygrain structures is a widely-accepted solution to achieve high hardness and high strength materials wherein dislocation pile-ups at the grain boundary resist subsequent plastic deformation^[1-2]. However, on the one hand further reduction of the grain size triggers notable grain boundary dominated softening, such as grain boundary migration, grain boundary decohesion which results in strength drop and limited tensile ductility^[3-4], on the other hand, the nanograined structure is thermally unstable and tends to undergo significant grain coarsening even at very low temperatures^[5]. This may also decrease the strength according to the Hall-Petch relation^[1]. Recently, nano-twinned polycrystalline metals that exhibit an unusual combination of ultrahigh strength and high ductility compared with conventional nanocrystalline materials have attracted intense interest^[6-16]. Many nanosized low-energy twin boundaries not only stabilize the nanograined structure but also could maintain a delicate balance between hindering dislocation motion and providing dislocation storage spacing^[13-14], thus to improve strength with few ductility tradeoff^[17]. Simulations and theoretical studies demonstrate that optimal yield strength and large ductility can be achieved by tuning the grain size and the twin spacing^{[11, 13-15]}. Therefore, to maintain the excellent mechanical properties of such materials, stabilization of nanotwinned structure with optimal grain size and twin spacing against grain growth is crucial.

Stability of the nanograined structures mainly depends on how to modify grain boundary characteristics. The approaches to stabilize the grain size are usually classified into two categories^[18-19]. The first is lowering the grain boundary energy by preferential solute segregation of the grain boundaries known as thermodynamic stabilization strategy^[20-24]. The second is decreasing the grain boundary mobility by solute drag, second-phase particle pinning, and chemical ordering, known as kinetic stabilization strategy^[25]. Besides the thermal effect, mechanical effect also influences the stability of the polygrain structure. Mechanically mediated grain growth has been observed in various nanocrystalline metals during deformation processes^[26-30]. The mechanical deformation can lead to grain refinement or coarsening, dependent on strain, strain rate, and strain gradient^[17]. In addition, atomistic simulations, dislocation-based theoretical models and experimental validations of stress-driven grain boundary migration, and grain-rotation induced coalescence of adjacent grains were proposed to reveal the mechanism of grain growth under mechanical deformation^[30-38]. Usually, twin induced grain growth is rarely reported in atomistic simulations although these studies show that nucleation of twin lamina can be promoted by emission of partial dislocation at the grain boundary or grain boundary dissociation especially in the case of low stacking fault energy^{[13, 39-41]}. In contrast, recent experimental studies report that the twinning processes lead to significant grain coarsening in nanotwinned Au films^[42-43] and Ni foils^{[16, 44]} under different mechanical loadings. So far, how the twinning processes influence the grain boundary migration and thus induce notable grain coarsening remains unclear.

In nanotwinned polygrain metals under the applied loading, the external elastic energy provides a thermodynamic driving force for the nucleation and growth of energetically favorable twin domains. When the twin starts to grow into adjacent grain, there are two options for the further motion of twins: twin transmission, namely, nucleation of another twin with different orientation in the adjacent grain or the continued twin growth after the adjacent grain shrinks by grain boundary migration. The latter provides the thermodynamic origin for notable twin-assisted grain growth. Because the twin domains and the grain texture coexist toward an energy-minimizing microstructure configuration, the coupled motions of twin boundaries and grain boundaries under the applied loading are required to minimize the energy cost which produces a domain-structure dependent driving force for their motions. It is interesting to note that the phase-field model for the arbitrary grain boundary motion^[45] and for the nucleation and growth of twins^[46-47] has been proposed separately. Here, we combine the two phase-field models together to study how the twinning processes assist grain growth. Similar phase-field approach has been successfully used to analyze the coupled motions of martensite twin boundaries and magnetic domain walls in magnetic shape memory alloys wherein complex domain processes result in intrigue magnetomechanical behavior of such magnetic shape memory alloys^[48].

In this work, we propose a phase-field model to analyze the coupled motions of twin boundaries and grain boundaries under the applied stress. The effects of applied stress amplitude and misorientation between adjacent grains on microstructural evolution in a twinned bi-crystal are revealed. Large misorientation angle and large stress amplitude favor the twinning-mediated grain boundary migration. The microstructural evolutions in twinned and un-twinned polycrystals under different applied stresses are simulated. The statistical analysis of the grain number with grain size in both polycrystals confirms the mechanism of the twin-assisted grain coarsening.

2 Phase-field model

In the phase-field model, we use two sets of long-range order (LRO) parameters to characterize the structural non-uniformities. As shown in Fig. 1, we define orientation field variables η_p (r) for grains with different orientations to describe a polycrystal structure^[45]. At the same time, we use another kind of LRO parameters to describe twin variants in the grains. For a face centered cubic (FCC) solid, there are 12 possible twin variants within a single crystal or a given grain in a polycrystal. For 2D simulations on the (110) plane, two modes of twin variants are included: one is along the [ 112] direction on the (111) habit plane, and the other is along the [1 1 2] direction on the (1 11) habit plane^[46]. The habit planes on the (110) plane of two modes are related by a rotation angle θ_twin =70.53°. For convenience, we define global coordinate axes (x, y) along [001] and [110] directions, respectively. Using the global coordinate axes as a reference, 48 orientation field variables η_p (r), p=1, 2, ..., 48, are used to characterize the grains with different orientations differentiated by rotation angles

respectively. In our calculation, we can set more orientation field variables to simulate more grain orientations in the polycrystals at the cost of the consumption of much computational time. However, it is shown that 48 possible grain orientations of the polycrystalline structure with the misorientation angle defined as

are enough to simulate the evolution of grain textures and growth kinetics^[45]. We define ϕ_{1, p} (r) and ϕ_{2, p} (r) to describe two twin variants in the grain η_p (r), p=1, 2, ..., 48, respectively. Each LRO parameter assumes a non-zero value within its corresponding phase and zero elsewhere. The total free energy of the system, containing local free energy, gradient energy, and long-range elastic interaction energy, is formulated as a function of these LRO parameters, whose variational derivatives with respect to these LRO parameters drive their evolutions over time governed by the time-dependent Ginzburg-Landau equations.

In our work, we focus on the microstructure evolution of coexisting twin domains and grain textures. Thus, we assume that the local free energy F_ch of the system is the superposition of the local free energies for grains and twins, respectively

(1)

where the former part represents local free energy for grains, the latter part represents local free energy for twins, and f_η, f_ϕ are defined as follows,

(2)

(3)

where Δf_η is the potential height between the two states of the minimum free energy of grains. The local free energy is written as the fourth-order Landau-type polynomial to protect convergence of the LRO parameters of grains during evolutions. The dimensionless coefficients A₂, A₄, A_γ are constants which are normalized so that the absolute minimum of f_η (η₁, η₂, ..., η₄₈) falls on the degenerate set of LRO parameters

Δf_ϕ is the potential height between the two states of the minimum free energy of twins. The local free energy is obtained by polynomial fitting based on the result of the first-principles simulation^[46]. The dimensionless coefficients B₂, B₄, B₆, B₈, B_γ are constants which are normalized so that the absolute minimum of f_ϕ (ϕ_{1, 1}, ϕ_{1, 2}, ..., ϕ_{1, 48}, ϕ_{2, 1}, ϕ_{2, 2}, ..., ϕ_{2, 48}) falls on the degenerate set of LRO parameters

Here, the gradient energy F_G of the system is used to characterize the energies from grain and twin boundaries

(4)

where the former part represents gradient energy for grains, and the gradient energy coefficient β_η is a constant due to the assumption of isotropic grain boundary energy, the latter part represents gradient energy for twins, and the gradient energy coefficients g_kl^{1, p} and g_kl^{2, p} are chosen for the two modes of twin variants in different orientation grains, respectively. To reflect the anisotropy of twin boundary energy, g_kl^{1, p}, g_kl^{2, p} can be written as in the global coordinate,

(5)

where a_kl^{R, p} are the elements of the axis rotation matrix defined as

(6)

and g_mn^{1, ref}, g_mn^{2, ref} are defined as follows:

(7)

(8)

where g₁₁ is much larger than g₂₂ to satisfy that the interfacial energy along the twin boundaries is much smaller than interface along other orientations.

In nanotwinned polygrain metals under the applied loading, each twin is linked to an eigenstrain which depends on the distributions of twins and grains. The eigenstrain, ε_kl⁰ (r, t) can be written as the function of the LRO parameter fields ϕ_{1, p} (r, t), ϕ_{2, p} (r, t), and η_p (r, t) by the equation,

(9)

where r and t are the position vector and the time, respectively, ε_kl^{twin1, p} and ε_kl^{twin2, p} are transformation stress-free strains of two twin phases in pth orientation grain, respectively,

(10)

where ε_mn^{1, ref}, ε_mn^{2, ref} are the components of the eigenstrain tensor of two twin modes in a single crystals^[46], respectively,

(11)

(12)

where ε₀ =γ_twin/2 and γ_twin is equal to , which is the magnitude of shear strains of fully twinned states for two modes along the [112] direction on the (111) habit plane and along the [1 1 2] direction on the (1 11) habit plane, respectively^[46]. According to the Khachaturyan elasticity theory, the elastic energy of an arbitrary coherent multiphase can be represented as a functional of the eigenstrain ε_kl⁰ (r) and the average strain ε_kl,

(13)

where c_ijkl is the elastic modulus tensor, and has a simple form for an elastically isotropic body as

(14)

where μ is the shear modulus, and ν is the Poisson's ratio. In Eq. (13),

is the Fourier transform of c_ijklε_kl⁰ (r), is a unit directional vector in the Fourier space, and

is a Green function tensor. The symbol ^* in Eq. (13) denotes the complex conjugate, and ∫_|k|≠0 is the integral in the Fourier space excluding the point |k|=0. Minimizing Eq. (13) with respect to ε_kl under the applied stress, σ_ij^ext, gives

(15)

where

Equations (1), (4), and (13) give the total free energy F of the system, which finally is a function of these LRO parameters,

(16)

The evolutions of those LRO parameters driven by spontaneous free energy minimization are governed by phenomenological time-dependent Ginzburg-Landau kinetic equations as follows:

(17)

(18a)

(18b)

where the variational derivatives, and , are the thermodynamic driving forces; L_η and L_ϕ are the kinetic coefficients of the movements of grain and twin boundaries, respectively. Substituting the total free energy function of LRO parameters into kinetic Eqs. (17), (18a), and (18b) leads to explicit non-linear kinetic equations fully describing the microstructure evolution during twin formation and grain growth, irrespective of the complexity of twin domains and grain textures.

In order to reveal how the key parameters control the microstructure evolution, we present the formulations in a dimensionless form for the numerical solutions of Eqs. (17), (18a), and (18b). All energy parameters in Eq. (17) are in units of the potential height, Δf_η, and in Eqs. (18a), (18b) are Δf_ϕ. The length Δx is measured in unit l, where l is the length of the computational grid increment. We employ dimensionless parameters in the following simulations:

(19)

The kinetic equations are solved numerically in the reciprocal space, and the dimensionless constants adopted from previous publications are listed in Table 1^[45-46].

In the present simulations, we firstly build an elastically homogeneous bicrystal model to reveal features of coupled evolutions of twin and grain boundaries since the grain-elasticity-driven and capillarity-driven grain boundary migration can be excluded. As shown in Fig. 2, the computational domain includes one mode of twin variants, which are along the [112] direction on the (111) habit plane, where η₁ =1, η₂ =1 represent grain 1 and grain 2, respectively, i.e., ϕ_{1, 1} =1, η₁ =1 represent twins in grain 1 and ϕ_{1, 2} =1, η₂ =1 represent twins in grain 2, . The computational cell consists of 256Δx × 256Δx grid sites. Secondly, we have performed simulations on evolutions of twin and grain boundaries in a polycrystal shown in Fig. 1, and the computational cell consists of 1 024Δx × 1 024Δx grid sites. In all the simulations, the periodic boundary condition is used.

3 Results and discussion

In a bicrystal shown in Fig. 2, the crystallographic axes of grain 1 and grain 2 are defined by rotation angles

(20)

The eigenstrains of twin 1 and twin 2 are ε_kl^{twin1, 1} and ε_kl^{twin1, 2}, respectively. Both the twins (twin 1 and twin 2) are of the same variant but in different grains to avoid any ambiguity. A set of simulations of the bicrystal with varying misorientation are carried out under a given shear stress σ^ext=0.15 μ ε₀, and a number of randomly positioned twin nuclei incorporated into bicrystal at initial stage. Evolutions of twins and grains boundaries in the bicrystal with the misorientation angle θ_mis =8°, 35°, and 45° are shown in Figs. 3-5, respectively. The misorientation angles are selected so as to demonstrate representative evolution paths of twins and grain boundaries. Figure 3 shows spatial-temporal evolution process of twins in Figs. 3(a)-3(d) and grains in Figs. 3(e)-3(h) in the case of θ_mis =8°. Twin 1 and twin 2 can simultaneously grow as a lamina shape first, extend in size within each grain and both coarsen gradually until both grains are almost completely twinned. Then the energetically favorable twin 1 further grows at the expense of shrinkage of twin 2 in accompany with the grain boundary migration.

In this case, twins can propagate from one grain to another, and the dominating driving force for twin-assisted grain boundary migration is the work done by the external loading σ_ij^ext (ε_ij^{twin1, 1} -ε_ij^{twin1, 2}). Since the misorientation angle is small, the driving force is weak, and the grain boundary migration is slow. The pathway of coupled motions of twin and grain boundaries with the features shown in Fig. 3 is called as mode Ⅰ.

Figure 4 shows spatio-temporal evolution process of twins in Figs. 4(a)–4(d) and grains in Figs. 4(e)–4(h) in the case of θ_mis =35°. It clearly shows that the probability of simultaneous growth of twin 2 known as twin transmission decreases with increasing misorientation angle, consistent with the previous report^[49]. Before grain 1 is fully twinned during twin 1 coarsening, the significant grain boundary migration via its bulging is observed near twin chain or twin band as shown in Figs. 4(b) and 4(f). In this case twins coarsen and grain boundary migration occur simultaneously. The grain boundary migration is found to be much faster than that in Fig. 3. The pathway of coupled motions of twin and grain boundaries with the feature shown in Fig. 4 is called as mode Ⅱ.

Figure 5 shows spatio-temporal evolution process of twins in Figs. 5(a)–5(d) and grains in Figs. 5(e)–5(h) in the case of θ_mis =45°. The results demonstrate that only energetically favorable twin 1 grows and then coarsens. The continued twin 1 growth can proceed after the grain 2 shrinks by grain boundary migration. The grain boundary migration driven by the bulging out of grain boundary occurs when the width of twin lamina is above a critical value. The bulging on the one hand facilitates energy releasing by the external loading σ_ij^ext ε_ij^{twin1, 1} ΔV with ΔV the volume change of the twin 1, on the other hand increases the grain and twin boundaries energy γ ΔS with γ the boundary energy and ΔS the area change of the twin 1. It is the energy competition that sets a critical value of the width of twin lamina. The formed serrated grain boundary by the bulging further speeds up the grain boundary migration, as shown in Figs. 5(g)–5(h). The pathway of coupled motions of twin and grain boundaries with the features shown in Fig. 5 is called mode Ⅲ.

In addition to the misorientation angle θ_mis of bicrystal, the applied stress σ^mis also influences pathways of coupled motions of twin and grain boundaries. Thus, we perform systematical simulations to investigate effects of the applied stress σ^mis and the angle θ_mis on evolutions of twins and grains boundaries in a bicrystal. The calculated phase diagram with respect to σ^ext and θ_mis shown in Fig. 6 demonstrates that three pathways for the coupled motions of twin and grain boundaries, named as mode Ⅰ, mode Ⅱ, and mode Ⅲ presented by different marks, respectively, can be changed by the applied loading and misorientation angle between adjacent grains. Since the fastest twinning-driven grain boundary migration is observed in mode Ⅲ, the result in Fig. 6 demonstrates large misorientation angle and large applied stress are in favor of twinning-driven grain boundary migration. Obviously, three pathways shown in Fig. 6 are obtained in the case of fixed grain boundary energy and twin boundary energy. Based on the analysis of energy variation, the grain and twin boundaries energies also influence the evolution paths which are not discussed here for simplicity.

To have a closer look at the mechanism of twin-assisted grain growth, we extend the simulation of microstructural evolution in a bicrystal to a polycrystal under applied shear stress

as shown in Fig. 1. We compare the resultant grain growth in the twinned and un-twinned polycrystals with the same initial grain texture. Figure 7 presents the statistical analysis of the grain number with grain size in the twinned and un-twinned polycrystals at the evolution time t^* =5 000. It clearly shows that the distribution range of the grain size in the twinned polycrystal is larger than that in the un-twinned polycrystal. The number of grain with larger grain size in the twinned polycrystal is more than that in the un-twinned polycrystal. This comparison confirms that the twinning process favors the grain boundary migration and accelerates grain growth, in good agreement with the experimental observations^{[16, 42-44]}.

Figure 8 shows coupled microstructure evolutions of twins and grains in twinned polycrystal in the case of Fig. 7. The black line represents the grain boundary. The red color represents twin variant along the [112] direction, and the blue color represents twin variant along the [1 1 2] direction. Three pathways for the coupled motions of twin and grain boundaries proposed in Figs. 3-5 can be observed as indicated by green, red, and yellow circle marks, respectively. When the misorientation angle between adjacent grains is large, their grain boundary bulges and migrates continuously, resulting in one grain growing and the other shrinking. The bulge out on the one hand induces notable grain boundary migration, on the other hand makes the grain boundary into new segments with high velocity due to a higher driving force.

The result in Fig. 6 indicates that high applied stress is in favor of mode Ⅲ which greatly enhances twin-assisted grain growth, so we further study the effect of the amplitude of applied stress on microstructure evolutions in both the twinned and un-twinned polycrystals. Figure 9 shows the distribution of grain size in the twinned and un-twinned polycrystals under different applied stress at the evolution time t^* =5 000. The difference of the grain size distribution between twinned and un-twinned polycrystals is not obvious when the applied stress is small. With the increasing of the applied stress, the difference becomes larger and the distribution of the grain size shifts toward large grain size, which demonstrates stronger twin-assisted grain growth. In the above simulations, we only consider the thermodynamic origin of twin-assisted grain growth where the grain and twin boundaries are assumed to be highly mobile. Figure 10 shows the microstructure evolution in a twinned polycrystal under applied stress in the limit of zero mobility of the grain boundary, namely, without grain boundary movement. It demonstrates that the energetically favorable twins can be both types of twin variants. They grow firstly within one grain, and then transmit into adjacent grains forming twin chains or twin bands. In some grains, grain boundaries with large misorientation angle block the twin transmission, and the twin cannot grow, which is different from the microstructure evolution shown in Fig. 8. This indicates that grain boundary mobility can also play an important role in twin-assisted grain growth.

In this work, our coarse-grained phase-field model for the coupled evolutions of grain and twin boundaries in nanotwinned polycrystals is based on the principle of minimizing the domain-structure dependent free energy. Although we have identified that the external elastic energy releasing is dependent on the applied loading level, and the misorientation angle between adjacent grains provides a thermodynamic driving force for twin-assisted grain growth, recent in-situ electron backscatter diffraction studies show that the interaction between twinning and grain boundary mobility is much more complex^[50]. In fact, the twinning process consisting of nucleation, propagation and lateral thickening depends on the local stress state along the twin boundary^[41]. For example, the twin nucleation is usually activated only when a threshold shear stress along the twin direction is overcome. A refined resolution of the local stress state needs one to take into account atomistic features of the grain and twin boundaries^[51]. An atomistically informed phase-field modeling of coupled evolutions of twin and grain boundaries needs to be developed in future studies. In addition, our model adopts an infinitesimal-strain based formulation to include the elastic energy in the free energy function and to solve the mechanical equilibrium equation. It is valid in the case of small eigenstrain. However, for the large eigenstrain introduced by twinning, a finite-strain formulation developed in future would be more accurate.

4 Conclusions

Inspired by the experimental observation of nanotwin-assisted grain growth under external loading, we develop a phase-field model capable of revealing the interactions between twining process and grain growth in the twinned polycrystals. The simulation results report three pathways for the coupled movements of grain and twin boundaries, dependent on the applied loading and orientation angle between adjacent grains. In particular, one of three pathways shows that the continued twin growth proceeds after the adjacent grain shrinks by grain boundary bulge and migration. This pathway provides insight into the mechanism of twin-assisted grain growth where the twinning process enhances grain boundary migration, dissociates grain boundary into segments and accelerates grain coalescence. Phase-field simulations indicate that the thermodynamic origin of twin-assisted grain growth is ascribed to the competition between the elastic energy and the interfacial energy. The results of our simulations further demonstrate that suppression of mode Ⅲ can effectively mitigate twin-assisted grain growth, which is beneficial to design nanotwinned metals with both high strength and good ductility.

Acknowledgements

We would like to thank Yujie WEI for helpful discussion and valuable suggestions.