Available online at www.sciencedirect.com Scripta Materialia 63 (2010) 1049–1052 www.elsevier.com/locate/scriptamat Analysis of the elastic strain energy driving force for grain boundary migration using phase field simulation Michael Tonks,a,* Paul Millett,a Wei Caib and Dieter Wolfc aFuel Modeling and Simulation, Idaho National Laboratory, P.O. Box 1625, Idaho Falls, ID 83415-3840, USA bDepartment of Mechanical Engineering, Stanford University, Stanford, CA 94305-4040, USA cCenter for Advanced Modeling and Simulation, Idaho National Laboratory, P.O. Box 1625, Idaho Falls, ID 83415-3840, USA Received 10 May 2010; revised 13 July 2010; accepted 23 July 2010 Available online 29 July 2010 We investigate elastic energy-driven grain boundary migration in a strained copper bicrystal using an atomistically informed phase field model. In a bicrystal experiencing a uniform strain, the softer grain has a lower energy density and grows at the expense of the harder grain. In a bicrystal experiencing heterogeneous strain, the softer grain has a higher energy density, yet it still grows. Our findings suggest that the softer grain will grow, irrespective of the difference in the energy densities. Published by Elsevier Ltd. on behalf of Acta Materialia Inc. Keywords: Grain boundary migration; Elastic deformation; Phase field model; Microstructure evolution Grain boundaries (GBs) migrate due to applied copper bicrystal under an applied strain similar to that driving forces. According to reaction-rate theory, the studied by Scho¨nfelder et al. [2]. Two cases are consid- GB velocity is typically a linear function of the driving ered: first, the case in which the strain is uniform force fd , i.e. v ¼ mfd ^r with m the GB mobility and ^r a throughout the bicrystal, and second, the case in which unit vector normal to the GB [1]. The driving force fd the strains are heterogeneous but the average strain in causes GB migration to lower the Gibbs free energy of the bicrystal is equal to the applied strain. The phase the system. The driving force can originate from various field model is validated by comparing to the results from sources, including GB curvature and inhomogeneities in Scho¨nfelder et al. [2]. Then we investigate the elastic the elastic strain energy in elastically anisotropic driving force by comparing the GB migration for the materials. two cases. In their molecular dynamics (MD) study, Scho¨nfelder We simulate GB migration in a two-dimensional et al. [2] investigated the elastic strain energy driving (2-D) copper bicrystal similar to the 3-D bicrystal used force. In order to obtain values for the GB mobility m in the MD simulations from Scho¨nfelder et al. [2]. Our at various temperatures, they analyzed GB migration 2-D bicrystal has the GB normal to the x-direction in a copper bicrystal with two flat high-angle twist and is homogeneous in the y-direction, while the 3-D GBs. Since the GBs had no curvature, a uniform defor- bicrystal from [2] has the GB normal to the z-direction mation was applied to the simulation cell resulting in a and is homogeneous in the x-andy-directions. In both uniform strain to drive GB migration. They measured bicrystals, the two grains are oriented at Æh=2 about the GB velocity for various deformation magnitudes the [001] axis, where h ¼ 43:60 (see the schematic in and found that the elastic driving force was equal to Fig. 1). Periodic boundary conditions are employed such that our 2-D bicrystal has two flat high-angle (001) sym- f ¼DE ð1Þ d el metric tilt boundaries, in contrast to the two high-angle where DEel is the difference in the elastic energy density (001) twist boundaries in the 3-D bicrystal from [2]. across the boundary. The bicrystal experiences the strain used by [2], In this work, we investigate the elastic driving force using a phase field grain growth model. We analyze a ¼ ðe1 e1 À e2 e2Þþðe1 e2 þ e2 e1Þð2Þ 2 where defines the magnitude of the strain, and grain 1 * Corresponding author. E-mail: [email protected] is softer than grain 2 with respect to this applied strain. 1359-6462/$ - see front matter Published by Elsevier Ltd. on behalf of Acta Materialia Inc. doi:10.1016/j.scriptamat.2010.07.034 1050 M. Tonks et al. / Scripta Materialia 63 (2010) 1049–1052 y heterogeneous strain is calculated at the start of a time step using the order parameter values from the previous time step. The calculated strain is then used to calculate Grain 1 Grain 2 the elastic energy density Eel for the phase field free en- 21.8°-21.8 ° ergy from Eq. (3). x The order parameters defining each grain are evolved with an Allen–Cahn equation according to GB 1 GB 2 GB 1 @/i @F @f L @C Figure 1. A schematic of our copper bicrystal with two high-angle tilt ¼L ¼L À Á ð6Þ boundaries, using periodic boundary conditions. @t @/i @/i 2 @/i where the order parameter mobility L ¼ 4m [4]. The 3wGB For the uniform strain case, the entire bicrystal is bicrystal is represented as a 1:28 lm  0:64 lm rectan- strained according to Eq. (2), resulting in the same en- gle. The system is solved using finite difference with a ergy densities within the grains as in the MD simulations 10 nm grid spacing and backward Euler time integra- from [2]. For the heterogeneous strain case, the average tion. A diffuse GB width of 4.5 grid spacings is used, strain is equal to Eq. (2) but the two grains accommo- since it was found to be the smallest value that did not date the applied strain differently. The xx and xy compo- alter the order parameter evolution. nents of the strain vary between the two grains, while the To investigate the behavior of the copper bicrystal, yy component is homogeneous due to compatibility. the GB migration is simulated at T ¼ 800 K for different The grain evolution is simulated with the phase field values of the applied strain magnitude . We first vali- model first presented by Fan and Chen [3], in which each date the model by comparing the behavior from the uni- form strain case to the MD results from Scho¨nfelder grain is represented by a continuous order parameter /i equal to one within the grain and zero in all other grains. et al. [2]. The GBs migrate such that grain 1 grows line- We also use expressions from Moelans et al. [4] to relate arly with time at the expense of grain 2. We determine each of the model parameters to the GB surface energy GB velocity from the slope of plots of the x-position of the GB centers with time (see Fig. 2a for an example). r, the GB mobility m and the diffuse GB width wGB.In our simulations, we employ the GB mobility and surface This calculation is repeated to calculate the GB velocity energy for pure copper that were determined using MD for each value of . The velocities calculated with our phase field model compare well with Eq. (1) from the by Scho¨nfelder et al. [2], i.e. r ¼ 0:708 J mÀ2 and MD results, as shown in Figure 2b. Our results also ÀQ m ¼ m0 exp , where kB is the Boltzmann constant, kBT show that grain 1 has a lower elastic energy density than À6 grain 2 (see Fig. 3a). T is the absolute temperature, m0 ¼ 2:5  10 1 For the heterogeneous strain case, grain 1 has a high- m4 JÀ sÀ1 and Q ¼ 0:23 eV. To include the effect of elastic deformation on the GB er elastic energy density (Fig. 3b). This occurs because grain 2 is stiffer and therefore accommodates less of migration, we add the elastic energy density Eel in the phase field free energy functional, i.e. the strain, resulting in a lower energy density. The strain Z distribution is uniform in both grains and changes smoothly but rapidly over the GB width. In spite of hav- F ¼ ðÞf ð/1; /2ÞþEel dV ð3Þ V ing a higher energy density, grain 1 still grows at the ex- pense of grain 2 for the heterogeneous strain case. As where f ð/1; /2Þ is the free energy density defined by grain 1 grows, the energy density in both grains de- Moelans et al. [4] and E 1 / ; / . The elastic el ¼ 2 ðCð 1 2Þ Þ creases, as does the energy density difference over the stiffness tensor C is a function of the order parameters boundary. The GB x-positions vary linearly with time defining the two grains according to and, again, we use the slope to determine the GB veloc- hð/1ÞC1 þ hð/2ÞC2 ity (see Fig. 4a for an example). The GB velocity varies Cð/1; /2Þ¼ ð4Þ hð/1Þþhð/2Þ where Ci is the elastic tensor from the corresponding grain orientation and the smooth function hð/iÞ¼ð1þ ab1.4 sinðpð/ À 1=2ÞÞÞ=2. Since copper has a cubic crystal i 2 structure, the elastic tensor is fully defined by three tem- 1.2 v = 2.4051 m/s perature-dependent elastic constants [2] 1 1 C11 ¼ 181:50 À :109T GPa 0.8 C12 ¼ 101:80 À :062T GPa ð5Þ v = 2.4553 m/s C ¼ 101:80 À :057T GPa 0 44 20 40 60 80 100 0 1 2 3 4 −4 For the heterogeneous strain case, the strain throughout x 10 the bicrystal is calculated with the spectral stress calcu- Figure 2.