0.3truein Grain boundaries are generally classified into two types, based on the misorientation angle () between the two half-crystals which form the grain boundary. Low-angle grain boundary () can be described by grain boundary dislocation model due to Read and Shockley [10], while the high-angle grain boundary () are generally modeled by coincident site lattice (CSL) [11]. High-angle grain boundaries are associated with higher grain boundary energy and are generally thought to promote GBS. In this work grain boundaries are modeled as planar and bicrystalline structures specified by CSL models. Though in reality grain boundaries are often curved interface, this model can be considered as a segment of the real" boundary. Three factors are generally used to describe CSL: the axis [uvw] about which the rotation occurs; the misorientation angle , and the reciprocal of the density of coincident sites in the (uvw) net, denoted as . As the consequence of the CSL model, the lowest-energy grain boundary structure for a given misorientation (characterized by ) is postulated to be the symmetrical configuration. Figure 2 illustrates the notation used in grain boundary designations throughout this work. In this figure, the rotation axis [uvw] is perpendicular to the plane of the paper (z-direction), consequently, the grain boundary plane (hkl) is x-y plane with y-direction aligned to the grain boundary normal; and the misorientation angle is computed from the two [001] directions of each of the bicrystals. Grain boundaries are designated as [uvw] N (hkl), thus the grain boundary shown in Figure 2 describes [001] (01) tilt boundary.
A series of CSL tilt grain boundaries about [110] with different grain boundary planes are
generated in this work.
Since the CSL tilt grain boundary is symmetric about the grain boundary plane (hkl),
which differs from the ideal crystal () only by the inversion of the stacking
sequence [4], its corresponding value is determined by:
takes value of 0.5 when (h2+k2+l2) is an even
number, and 1 when (h2+k2+l2)
is an odd number. It should be noted that in cubic lattice can
take on only odd values.
In this work, the boundary rotation axis [uvw] is always set to [110],
and will not be included in
the boundary description whenever possible. The 17 tilt [110] CSL boundaries studied
in this work are as follows: (11),
(12),
(21), (14),
(13), (32),
(34),
(31),
(15), (52),
(25), (41), (18),
(43),
(56), (35) and (51).
The misorientation angle is computed as twice the angle between the directions of x-axis
and [001] direction in any one of the bicrystal, and is shown besides the boundary description.
For each of the CSL boundaries, the computational crystal was generated based on the orientation of a given grain and the symmetry between that and the adjacent grain across the boundary plane. During construction of the initial unrelaxed structures, it may be necessary to obtain lower-energy states by removing (or adding) atoms from the boundary plane. Since grain boundaries are extended defects in two dimensions, but inhomogeneous in the direction normal to the grain boundary plane, it is usual to construct a computational crystal that is periodic only in 2-D plane of the interface (x- and z- directions in this work). In the grain boundary normal direction (y-direction), free-surface boundary conditions are imposed. Consequently, the crystals are designed to be large enough in y-direction to remove the free surface effects on the grain boundary structure. The computational crystals used in this work contain about three to five thousand atoms.
Figure 3 shows the final equilibrium structures of selected grain boundaries obtained using molecular statics simulations. It should be noted that only a portion of the whole computational crystal close to the grain boundary is shown. The open and filled circles represent atoms in two adjacent (110) atomic layers, which have been projected in a plane normal to z=[110] direction. It is observed that during the simulation process, most of the atomic movement occurs near the grain boundary plane (in the relaxed state) compared to the initial unrelaxed configurations. The equilibrium configurations shown in Figure 3 were compared to other available computational and experimental results. The key aspect to be compared is the micro-facet structural details near the boundary which includes the relative positions of atoms in the boundary and neighboring planes. The present results for two of the CSL structures (21) and (13) agree well with those available in the literature [1, 12]. Also the present results agree for [21] with the experimentally observed tilt boundaries using high-resolution transmission electron microscopy (HRTEM) [7]. These comparisons not only validate the EAM potentials functions and the computational methods used in this work, but also endorse the prediction of other CSL boundary structures shown in Figure 3.
The distribution of energy across the grain boundaries corresponding to Figure 3 is next computed. Figure 4 shows the energy associated with atoms as a function of distance from the grain boundary plane (x-axis zero being at the grain boundary). The energy increases as the grain boundary is approached from either side of the bicrystal. There is significant variation in the energy levels for the different boundaries considered. The width of the grain boundary can be defined when the energy of atoms equals to the value of energy in a perfect crystal (-3.58 eV for aluminum). By this definition, the width of grain boundaries varies with different boundary structures (see Figure 4), from a maximum 10 to almost zero in (11) structure.
The energy of the individual atoms (plotted in Figure 4) can be used in the evaluation of grain boundary energy which represents the energy per unit area of grain boundary plane. This grain boundary energy is equal to the energy of atoms within the width of the grain boundary in the defective system less than that for the perfect crystal, divided by the area of the grain boundary plane. Figure 5 (a) shows the grain boundary energy (Egb) for all the tilt grain boundaries studied in this work, plotted as a function of the misorientation angle . As can be seen from the plot, three energy cusps for special" angles are observed in this work, which correspond to three twin boundaries: (11), (12) and (13). Wolf [4] and Hasson et al. [6] have conducted similar grain boundary simulations for copper and aluminum using pair potentials. However, they only observed energy cusps for the (11) and (13) orientations but not for the (12) orientation. Since (12) is also a twin boundary it is reasonable to expect the (12) boundary to be also a low-energy defect. The observation of the new low energy configuration (12) can be ascribed to the use of EAM potentials in this work compared to the pair potentials used in the earlier works. In the EAM calculations, the configuration energy is composed of a simple pair interaction term plus an embedding" function which specifies the dependence of energy on local coordination. The ability to treat deviations in local coordination has been shown to be crucial in obtaining reasonable agreement with the relaxation at interfaces and free surfaces. Atoms near the boundary interface have different environment from the atoms far away from the interface. Thus atoms interacting across the interface experience an electron density different from that of atoms interacting with each other on the same side of the interface. This intrinsically anisotropic character of the atoms near the interface is not taken into account by any pair potential.
Twin structures provide coherent boundaries in the interface of two-half crystals and result in low-energy defects. They are characterized by very low energies as shown in Figure 5. In contrast, other grain boundaries provide large mismatch between atoms close to the grain boundary plane and therefore have higher grain boundary energies. In order to quantify the deviation of grain boundary structures from perfect FCC arrangement, the radius distribution function (RDF) for typical structures were computed. In perfect FCC crystal (), the squared ratios of the n-th neighbor distance and the first nearest neighbor distance of an atom take integer numbers (equal to n), and the number of neighbors at each neighbor distance is defined, as shown in Figure 6 (a). In the crystals with grain boundary defects, the number of neighbors at each neighbor distance is averaged from the four atoms in one primitive CSL cell, with two of them residing at the boundary interface and other two residing in one side of the bicrystal. As can be seen from figures 6 (b)-(d), (11) boundary almost maintains the same neighbor arrangement as the perfect crystal, which results in the lowest grain-boundary energy state. (13) boundary has less deviations from the perfect arrangement than (21), which is consistent with their corresponding grain-boundary energy values.
The most important parameter which controls the grain boundary energy is
the
interplanar spacing of the grain boundary plane [4], which, in cubic Bravais lattice
is given by
where takes value of 0.5 or 1 (base on the details of
geometry [4]),
and ae is the lattice parameter.
When creating a planar defect on a given crystallographic plane (hkl)
the perfect stacking of lattice in its normal direction is destroyed.
For example, all atoms facing each other across the grain boundary
are forced to come closer when the defect is created, and they substantially repel
each other.
The grain boundary energy is hence expected to increase with the decreasing
separation of the lattice planes facing each other
across the grain boundary plane.
Figure 5 (b) gives the interplanar spacing for each boundary and Figure 5 (c) shows
its relation with grain boundary energy. It is seen
that large d-spacing gives rise to lower grain boundary energy.
Since the interplanar spacing is proportional to planar atom
density, the close-packed planes are the most favored lattice planes for the
accommodation of low-energy symmetric grain boundaries.