The contrasting fracture behaviour of twin boundaries and general boundaries – A first principles study based on experimental observation

Abstract

Experimental observations of grain boundary fracture behavior have been used as exemplars, and boundaries with the same crystallography as the experiments were developed. These boundaries were non-symmetrical, and this necessitated the first ever study of non-symmetrical boundaries in magnesium using density functional theory. The broad agreement of the calculated boundary cohesion values with the experimental observations of fracture behavior showed the simulations to be good approximations of real behavior, and from this point the simulations were further interrogated to understand the differences between the boundary types and solute species. Solutes with both larger and smaller radii than magnesium had a preference for segregation to the grain boundary. The boundary cohesion was examined by the parameter known as embrittlement potency, and it was found that solutes smaller than magnesium had a toughening effect, while those solutes larger than magnesium had more tendency to embrittle. The two boundaries studied in most detail, the $\left\{10\bar{1}2\right\}$ twin boundary, and a general grain boundary observed experimentally, showed different cohesive behaviors. Although the symmetrical twin boundary behaved in a similar manner to previous reports, the non-symmetrical boundary showed more complex cohesive behavior, highlighting the importance of studying the non-symmetrical boundaries that predominate real materials.

Introduction

In metallic alloys, grain boundaries govern a plethora of properties and behaviors such as strength [1], toughness [2], and corrosion susceptibility [3]. Understanding the energetics and the atomic-scale behavior of grain boundaries in engineering alloys is therefore of significant interest. To expand our understanding of grain boundary structure and obtain the corresponding energetic cost of creating them, atomistic simulations have been extensively used [[4], [5], [6]] and in particular, a large set of symmetrical grain boundaries in HCP magnesium with tilt axes $\left[1\bar{2}10\right]$ and $\left[0\bar{1}10\right]$, was simulated by molecular dynamics (MD) [7,8]. The calculated grain boundary energies as a function of misorientation angle showed four local minima.

Ab initio simulation revealed a large driving force for segregation of Na and Zn solutes to a symmetrical boundary in Al whereas its mechanical strength was computed to be higher for the former and lower for the latter [9]. Also, while Cu segregation driving force to a grain boundary in Al is rather large and leads to its cohesion, Mg segregation driving force is smaller and results in its embrittlement [10]. The segregation of six transition metals from both 5th and 6th rows of the periodic table to eight symmetrical boundaries in W were studied and the cohesiveness of segregated boundaries showed a similar shape with increasing atomic number [11].

Ab initio calculations can also be used to assess electronic structure and bonding characteristics of grain boundaries and solute segregation [12]. Yamaguchi and coworkers [2] computed grain boundary cohesion and the distribution of charge density in Mg, Mg–Ca, Mg–Ca–Zn and Mg–Ca–Al alloys. The increased fracture toughness of Mg due to the addition of Ca and Zn or Al was attributed to the higher charge density observed in the GB region. In a study of interfacial fracture in Mg alloys an electronic structure-based approach was applied [13] and three twin boundaries (TB) were considered to be segregated by a dozen of different solutes. Most solutes showed tendency to segregate into grain boundaries to reduce strain energy. It is pertinent to note that phase field simulation has also been applied to model transformations between martensitic variants and multiple twinning within martensitic variants [14] prescribing twin interface energy and width. In another study, a thermodynamic potential was developed for temperature and stress induced transformation between multiple phases [15].

While experimental quantification of grain boundaries is relatively routine using techniques such as electron backscattered diffraction or transmission electron microscopy (TEM), examination of grain boundary behaviors through computational methods is not as easy. To the best of the authors’ knowledge, first principles simulations have been restricted to symmetrical boundaries [2,13,16,17] which can be calculated using simulation cells with a relatively small number of atoms. However, in real materials, perfectly symmetrical boundaries are rarely observed, with the vast majority of boundaries exhibiting irrational crystal rotations on high order planes [[18], [19], [20]]. Simulations of these real boundaries necessitate simulation cells with a large number of atoms due to the requirement for periodic boundary conditions and are therefore computationally very expensive. These types of “real” boundaries have been examined using molecular dynamics approached in the past but have not before been examined using density functional theory (DFT) due to their size. One significant advantage of first principles approaches over others is the ability to use different element chemistries without the need to develop new inter-atomic potentials. For this reason, segregation of solutes to boundaries in dilute alloys is best studied by the first principles approach. Despite the large simulation cell sizes required, simulation of real (irrational) boundaries with DFT is possible, and for the first time two such boundaries are examined in the present work.

A recent publication described direct experimental observation of the contrasting fracture behavior of different boundary types using in-situ tensile TEM [21]. It was observed that in pure Mg, cracks propagated along general grain boundaries, and were unaltered by the formation of $\left\{10\bar{1}2\right\}$ deformation twins ahead of the crack tip. This behaviour changed markedly by the addition of Y to the alloy, where the cracks showed a preference for propagation along the twin boundary rather than the grain boundary. In another study, macroscopic fracture surfaces of Mg–Zn showed a preference for fracture at the twin-matrix interface, while in samples of Mg–Ca cracks preferred growth along grain boundaries rather than twin boundaries [22]. So, we can see that the presence of solute markedly effects the fracture behaviour of magnesium alloys, but we are yet to understand why. Most importantly, the twin boundaries and general grain boundaries behaved differently in these studies, and it is not yet known why these boundaries exhibit different behaviors depending on their crystallography. This gap in knowledge stems largely from the difficulties in creating non-symmetric boundary simulations, and the expensive computation time of such large cell sizes.

First principles studies of grain boundary cohesion (vis-à-vis fracture behavior) are relatively straightforward, and many examples on symmetrical tilt boundaries can be found, for example, [2,13]. The bonding at the boundary can be described as the sum of three different factors: bond-breaking, chemical interactions, and atomic size effects. A comprehensive literature review of all systems by Gibson and Schuh [23] showed that the boundary cohesion is predominantly determined by the bond breaking factor, above the other two. Although this general statement may be true for the symmetrical tilt boundaries that are usually studied, the behavior of complex irrational boundaries has not before been examined. Therefore, in the present work the contributions to grain boundary cohesion are assessed on complex non-symmetric boundaries. In doing so, an extensive comparison between twin and general grain boundaries is presented.

This paper is split into two parts, the first part compares the experimental and theoretical results, providing robust evidence that the first principles calculations correlate with what is seen experimentally. Once the foundations for the work have been set, the second part of the paper extends this methodology to examine the behavior of 19 different segregants in order to create an alloy design framework for fracture toughness in magnesium alloys.