We have performed ab-initio tensile tests of bulk Al along different tensile axes, as well as perpendicular to different grain boundaries to determine mechanical properties such as interface energy, work of separation and theoretical strength. We show that all the different investigated geometries exhibit energy-displacement curves that can be brought into coincidence in the spirit of the well known universal binding energy relationship curve. This simplifies significantly the calculation of ab-initio tensile strengths for the whole parameter space of grain boundaries.
The prediction of the mechanical properties of polycrystals requires knowledge about the mechanical properties of all interfaces, i.e. grain boundaries, in their microstructure.
So far, most mesoscale models of microstructure-property relationships as used in continuum simulations of deformation and fracture make rather simple assumptions for the variation in grain boundary properties with the boundary geometry 1,2 . With ab initio electronic structure calculations it is possible to determine the cohesive energy, elastic modulus, sliding barrier, and theoretical strength of interfaces accurately and quantitatively. The considerable computational effort, in comparison e.g. to atomistic simulations employing empirical potentials, is easily feasible with modern computers if the investigations are restricted to grain boundary structures based on coincidence-site lattices. The ab initio approach is desirable to avoid problems of transferability of empirical potentials, usually fitted to reproduce equilibrium properties, to non-equilibrium processes such as failure. It is indispensable whenever a phenomenon is controlled by the electronic structure. This is the case in systems with directional bonds, or when studying the influence of chemical composition, and alloying effects. Nevertheless, sampling the five parameter space given by the geometric degrees of freedom of the grain boundary (rotation axis and angle, and the grain boundary normal 3 ) by ab initio calculations remains a challenge. Models exist that relate the energies of certain subsets of this space to the grain boundary geometry. Most well known is the dislocation model for small angle tilt grain boundaries based on the picture of Read and Shockley, which can be extended empirically to large-angle tilt grain boundaries 4 . If the energy of a grain boundary can be related to its geometric parameters via such models, the description of the energy hypersurface is significantly simplified. However, so far it seems that there is no such correlation which would be valid in the complete parameter space 5,6 . The complexity of the problem is increased by the fact that for use in mesoscale models we are not only looking for a function that describes the grain boundary energy as function of misorientation, but also for its first and second derivatives with respect to a displacement from the equilibrium volume, i.e. the maximum stress and the elastic modulus. Nevertheless, the situation is not hopeless, as the analytic function of this energy-displacement curve is the same for any grain boundary geometry, if obtained under the same loading conditions. The demonstration of this fact, a universal binding behaviour in the spirit of Rose’s universal binding energy relationship (UBER) 7 , is subject matter of this paper.
In section II we describe our computational procedure, including an explanation of grain boundary nomenclature and details on different ways to perform “ab initio” tensile tests. In section III we review the UBER and its implications. In the results’ section, sec. IV, we give details about the grain boundary structures after a full optimization of the microscopic degrees of freedom (IV A) and the corresponding energies (IV B). The results of the tensile tests are presented in section IV C. The universal elastic behaviour under tensile load of all systems investigated is demonstrated in section IV D, and the relationship between energies and strength is discussed in section IV E. We summarize our insights in section V.
For a given orientation of the tensile axis, we constructed supercells for bulk, surface, and grain boundary calculations of the same size and shape. In other words, starting from a bulk supercell containing N atomic layers, half of the planes were replaced by vacuum to create a surface slab, or half of the planes were replaced by the same number, but with a misorientation, to create a grain boundary structure.
In detail, supercells for tensile tests were constructed for Al bulk, such that the z-axis is oriented along the [111], [112], [113], and [114] direction, i.e. defining (111), (112), ( 113) and (114) as the cleavage plane. In addition we constructed special grain boundaries containing these planes as grain boundary planes. These were the Σ3 (111) [111] In this nomenclature the use of Σ indicates that for the chosen misorientation a periodic superstructure can be found, the so-called coincidence site lattice (CSL). Therefore these grain boundaries are also called special grain boundaries. The value of Σ is the volume of a unit of this CSL divided by the volume of the cubic Al unit cell, i.e. it is a measure for the periodicity of the grain boundary. The grain boundary plane is given in round brackets, the direction of the axis of misorientation in rectangular ones. All tilt grain boundaries considered here are symmetrical, which means that the grain boundary plane divides the misorientation angle in two equal parts. In othe
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