Elsevier

Acta Materialia

Volume 65, 15 February 2014, Pages 19-31
Acta Materialia

Capillary-driven grain boundary motion and grain rotation in a tricrystal: A molecular dynamics study

https://doi.org/10.1016/j.actamat.2013.11.059Get rights and content

Abstract

We report on molecular dynamics (MD) simulations of a tricrystal composed of a cylindrical grain embedded at the center of a plane grain boundary (GB). The embedded grain shrinks by capillary forces and eventually vanishes. This process is often accompanied by rotation of the embedded grain in either a clockwise or counter-clockwise direction. Using the geometric theory of coupling between GB motion and grain translations, we propose a model capable of predicting the direction of the grain rotation depending on the crystallographic parameters of the three grains. Full agreement has been found between the model predictions and the MD simulation results for both spontaneous grain shrinkage and in the presence of applied shear stresses. The consequences of these results for grain rotation in polycrystalline materials and possible extensions of the model to multiple grains are discussed.

Introduction

Grain rotation is part of microstructure evolution in polycrystalline materials and has been observed experimentally during plastic deformation [1], [2], recrystallization [3] and grain growth [4], [5]. Rotation of an isolated cylindrical grain was predicted theoretically [6], [7], [8] and observed in computer simulations by molecular dynamics (MD) [6], [9], [10], [11], [12] and the phase-field crystal (PFC) method [13]. In most cases the grain rotation could be explained by the existence of coupling between grain boundary (GB) motion and shear deformation of the lattice [6], [7], [8], [14], [15].

In real materials, isolated grains are extremely rare. Most grains are surrounded by several other grains and are separated from those by multiple GBs and triple junctions. One could expect that this complexity would suppress the grain rotation, or at least make it a rare occurrence, for at least two reasons. First, the triple lines can be expected to act as pinning sites since the grain rotation requires accommodation of incompatibilities along the triple lines [16], [17], [18]. Second, assuming that the magnitude and sign of the grain rotation depend on geometric parameters of the GB, different GBs would try to rotate the grain in different directions, reducing or even blocking the net rotation. Nevertheless, in certain cases the individual GBs can exert forces acting in the same direction and collectively causing the grain to rotate. The ability to predict such cases requires a better understanding of the geometric conditions for grain rotation by individual GBs and the role of triple junctions in this process.

Because of the complexity of the general polycrystalline problem, it is strategically meaningful to start with a relatively simple configuration amenable to quantitative analysis and then continue to build up the complexity by adding more grains. As such, a perfect starting point is offered by a tricrystal consisting of a cylindrical grain sitting on a plane GB. This configuration involves two different GBs surrounding the embedded grain and two different triple lines. The embedded grain spontaneously shrinks by capillary forces and may or may not rotate in the process.

This tricrystalline structure was studied by MD simulations in a two-dimensional Lennard–Jones system with a triangular lattice [11]. No rotation was found when the two curved GBs were symmetrically equivalent, leading to cancellation of the rotational forces. When the symmetry was broken, rotation was observed in some cases but not in others, and was apparently influenced by the presence of open surfaces in the simulated models. The important observation was, however, that the presence of two different GBs and two triple lines did not prevent the grain rotation. More recently, Wu and Voorhees [13] reported on two-dimensional PFC simulations of a similar tricrystalline system with hexagonal lattices. The geometry was symmetrical with the curved GBs having opposing misorientations of ±5.2°. As expected from this symmetry, no grain rotation was observed, in agreement with the MD simulations [11]. Although interesting insights into the dislocation mechanisms of GB migration were obtained, grain rotation under asymmetric conditions was not tested.

In this work we address the same tricrystalline configuration but in a more systematic manner. Based on the analysis of geometric coupling factors, we propose a simple analytical model that permits predictions of the direction of grain rotation, depending on the misorientation angles of the GBs. We then conduct a series of MD simulations designed to test this model. All three GBs are chosen to be [0 0 1] tilt type. The atomic interactions are modeled with an accurate atomistic potential to ensure realistic character of the results. As a further test of the model, we conduct simulations under a shear stress applied parallel to the plane GB and driving the collective motion of all three boundaries.

In Sections 2 Coupling model for curved grain boundaries, 3 Effective coupling factor and rotation of an embedded grain we introduce our model and propose analytical expressions for the effective coupling factor governing the grain rotation. These two sections also serve to introduce a set of sign conventions for the angles and coupling factors; the nature of the problem makes it important to adopt and strictly follow such conventions in order to avoid confusion. In Section 4 we describe the simulation methodology and details of the GB geometries tested in this work. The results of the MD simulations for the spontaneous shrinkage of the embedded grain are presented in Section 5, followed by stress driven simulations in Section 6. We conclude the paper by discussing the comparison between the proposed model and the simulation results and outlining future work (Section 7).

Section snippets

Coupling model for curved grain boundaries

To set the stage for the discussion of curved GBs, we will first review the geometric theory of coupling for plane GBs. Fig. 1 illustrates the geometry of a plane asymmetrical [0 0 1] tilt GB between two face-centered cubic (fcc) crystals, defining the tilt angle θ and the inclination angle ϕ. Angle θ measures the misorientation between [100] directions in the grains, with θ=0 for a single crystal and θ>0 when the [100]U axis in the upper grain is rotated counter-clockwise relative to [100]L in

Effective coupling factor and rotation of an embedded grain

As a test of the proposed model, we first apply it to the case of an isolated cylindrical grain with a circular cross-section shrinking by capillary forces [6], [7], [11], [12], [13]. The dynamics of the shrinking process are described by the equations [7], [12]vn=-Ṙ=Mγ/Rv||=Rθ̇=βvnwhere R is the grain radius, γ is the GB free energy, which is assumed to be constant, M is the GB mobility coefficient and the dot denotes the time derivative. A positive coupling factor causes the grain to rotate

Methodology of atomistic simulations

Copper was chosen as the model material to enable comparison with previous simulation studies of coupled GB motion and grain rotation [8], [12], [14], [15], [19], [20], [27]. Atomic interactions in Cu were modeled with an embedded-atom potential fit to experimental and first-principles data [28]. The potential predicts that the melting temperature of Tm=1327K in good agreement with the experimental value of 1356 K. The MD simulations employed the ITAP Molecular Dynamics (IMD) program [29]

MD results for spontaneous grain shrinkage

At all temperatures and geometries tested, the embedded grain shrank spontaneously until the simulation block became a bicrystal. In most cases the remaining GB was flat. In some cases a kink was observed immediately after the embedded grain had vanished. We expect that, given enough time, the kink would disappear, but a proof would waste computational resources and was not of interest in this study.

Although the initial geometry had a 180° triple junction angle, as shown in Fig. 5, during the

MD simulations under applied shear

The stress-driven simulations were conducted on the embedded grain with θU=16.26° and θL=73.74°, corresponding to case 1 in Table 1. The low-angle grain is sitting on a high-angle GB with θp=57.48°, as shown in Fig. 7. Recall that, by the crystal symmetry, βe=0 and the embedded grain spontaneously shrinks without rotation.

The applied shear stress breaks the symmetry and is expected to induce both GB motion and grain rotation. Specifically, for the lower curved GB we have βL<0 and expect its

Discussion and conclusions

We have investigated the process of capillary-driven shrinkage of an embedded cylindrical grain initially centered on a plane GB. Although this tricrystalline structure is highly idealistic, it captures some of the key features of polycrystalline materials: namely, the multiplicity of GBs surrounding a given grain and the presence of triple junctions. This structure is ideal for an initial study of GB motion and grain rotation in the presence of multiple GBs and triple junctions.

We find that

Acknowledgements

This work was supported by the U.S. Department of Energy, the Physical Behavior of Materials Program, through Grant No. DE-FG02-01ER45871.

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