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Abstract
The zero-temperature energies and equilibrium volume expansions of point-defect-free grain boundaries (GBs) on the two densest planes of b.c.c. metals have been determined using a Finnis—Sinclair potential for Mo and Johnson's potential for α-Fe. The energies and volume expansions of the (100) boundaries are found to be about two to three times larger than those of the (110) boundaries. A close correlation between large volume expansion and high GB energy is observed. Since both potentials yield qualitatively very similar results it is concluded that the short-range repulsion between atoms is the dominating contribution to the energy in the GB region and therefore, electronic many-body effects arising from a local volume dependence of the interatomic interactions are relatively unimportant.
... In the last few decades, a large number of GBs in bcc-Fe have been characterized within the framework of interatomic potentials [18][19][20][21][22]. However, to gain a better understanding of the GB structure-property correlation, an accurate description of local bonding across GBs is required [23], and it is necessary to go beyond interatomic potentials. ...
... According to our TB calculations, Σ11 and Σ33 twist GBs are quite close in energy (table 4). We thereby confirm previous calculations which show that all the (110) GBs have a similar energy except for the cusp at 3 S misorientation [18,59]. It can be seen that (110) twist GBs exhibit much lower energy than all the STGBs except for Σ3(112). ...
... Therefore, it is not surprising to see a high fraction of (110) boundaries in the lath martensite [14] owing to low interface energy. Interestingly, similar to (110) planes in bcc structure, (111) planes are also the densest planes and have the lowest twist GB energies in fcc structure [18,60]. Corre- spondingly, (111) g and (110) a¢ are parallel in the K-S orientation relationship and the other two common orientation relationships, namely, the Greninger-Troiano and Nishiyama- Wasserman relationship. ...
Grain boundaries (GBs) have a significant influence on material properties. In the present paper, we calculate the energies of eleven low-σ (σ > 13) symmetrical tilt GBs and two twist GBs in ferromagnetic bcc iron using firstprinciples density functional theory (DFT) calculations. The results demonstrate the importance of a sufficient sampling of initial rigid body translations in all three directions. We show that the relative GB energies can be xplained by the miscoordination of atoms at the GB region. While the main features of the studied GB structures were captured by previous empirical interatomic potential calculations, it is shown that the absolute values of GB energies calculated were substantially underestimated. Based on DFT-calculated GB structures and energies, we construct a new d-band orthogonal tight-binding (TB) model for bcc iron. The TB model is validated by its predictive power on all the studied GBs. We apply the TB model to block boundaries in lath martensite and demonstrate that the experimentally observed GB character distribution can be explained from the viewpoint of interface energy.
... The literature is rich with MD data on a large subset of the five-dimensional configuration space of interfaces. Data for FCC-FCC (Wolf, 1989a(Wolf, , 1990aSchmidt et al., 1998;Wolf et al., 1992;Merkle and Wolf, 1992;Tschopp and McDowell, 2007a,b), BCC-BCC (Wolf, 1990c(Wolf, , 1989b(Wolf, , 1991, HCP-HCP (Wand and Beyerlein, 2012a, b), and FCC-BCC (Kang et al., 2012b) provide solid benchmarks for model verification. ...
... Fig. 9. BCC STwGB. MD data from Wolf (1989bWolf ( , 1991. ...
... The results for the model for STwGB boundaries are compared against MD data from Wolf (1989bWolf ( , 1991 in Fig. 9. For the {001} STwGB we notice close agreement with the MD data, matching the minor cusps at θ =°°37 , 53 . ...
We develop an explicit model for the interfacial energy in crystals that emphasizes the geometric origin of the cusps in the energy profile. We start by formulating a general class of interatomic energies that are reference-configuration-free but explicitly incorporate the lattice geometry of the ground state. In particular, away from the interface the energy is minimized by a perfect lattice. We build these attributes into the energy by locally matching, as best as possible, a perfect lattice to the atomic positions and then quantifying the local energy in terms of the inevitable remaining mismatch, hence the term lattice-matching used to describe the resulting interatomic energy. Based on this general energy, we formulate a simpler rigid-lattice model in which the atomic positions on both sides of the interface coincide with perfect, but misoriented, lattices. In addition, we restrict the lattice-matching operation to a binary choice between the perfect lattices on both sides of the interface. Finally, we prove an on the interatomic energy and use that bound as a basis for comparison with experiment. We specifically consider symmetric tilt grain boundaries (STGB), symmetric twist grain boundaries (STwGB) and asymmetric twist grain boundaries (ATwGB) in face-centered cubic (FCC) and body-centered cubic (BCC) crystals. Two or more materials are considered for each choice of crystal structure and boundary class, with the choice of materials conditioned by the availability of molecular dynamics data. Despite the approximations made, we find very good overall agreement between the predicted interfacial energy structure and that calculated by molecular dynamics. In particular, the positions of the cusps are predicted well, and therefore, although surface reconstruction and faceting are not included in the model, the dominant orientations of the facets are correctly predicted by our geometrical model.
... In addition to characteristic dependencies on n, the GB properties discussed above may also be correlated to each other. For example, several authors have suggested that high GB specific excess volumes are correlated with high GB energies [66,[74][75][76][77][78][79][80][81]. Figure 3 plots c GB of all GB structures investigated here against their Dv GB values. We find that minimum GB energies-indicated by diamond symbols in Fig. 3-increase monotonically with specific excess volume. ...
... Even in the R11 GB, where both c GB and Dv GB exhibit periodic variations with n, these two quantities are not correlated with each other. Thus, the relationship between c GB and Dv GB discussed in Ref. [66,[74][75][76][77][78][79][80][81] should be understood to apply to comparisons between different GBs in their minimum energy states, not to comparisons between different states of the same GB or between different GBs driven far from equilibrium. Figure 3 also shows that the GBs that exhibit large fluctuations in c GB upon continuous vacancy loading also exhibit large fluctuations in Dv GB . ...
... The introduction of vacancies also changes a GB's specific excess volume and GB stress. Several investigators have argued for a monotonic relation between GB energy and excess volume [66,[74][75][76][77][78][79][80][81]. Our study adds an important qualification to this view by showing that it applies only to comparisons between different GBs in their minimum energy states, not to comparisons between different states of the same GB or between different GBs driven far from equilibrium. ...
We use atomistic modeling to study the response of three non-coherent grain boundaries (GBs) in Cu to continuous loading with vacancies. Our simulations yield insights into the structure and properties of these boundaries both near and far from thermal equilibrium. We find that GB energies vary periodically as a function of the number of vacancies introduced. Each GB has a characteristic minimum energy state that recurs during continuous vacancy loading, but in general cannot be reached without removing atoms from the boundary. There is no clear correlation of GB energies with GB specific excess volumes or stresses during vacancy loading. However, GB stresses increase monotonically with specific excess volumes. Continuous vacancy loading gives rise to GB migration and shearing, despite the absence of applied loads. Successive vacancies introduced into some of the boundaries accumulate at the cores of what appear to be generalized vacancy dislocation loops. We discuss the implications of these findings for our understanding of grain boundary sink efficiencies under light ion irradiation.
... There have been a number of atomistic simulations of grain boundary energies in bcc metals [19][20][21][22][23][24][25]. Wolf [22][23][24] showed that the energy anisotropies of Fe and Mo were similar for symmetrical tilt boundaries, twist boundaries on (1 0 0) and (1 1 0) planes, and certain general grain boundaries. ...
... There have been a number of atomistic simulations of grain boundary energies in bcc metals [19][20][21][22][23][24][25]. Wolf [22][23][24] showed that the energy anisotropies of Fe and Mo were similar for symmetrical tilt boundaries, twist boundaries on (1 0 0) and (1 1 0) planes, and certain general grain boundaries. Morita and Nakashima [20] investigated the boundary energy of <0 0 1> symmetric tilt boundaries in Mo, producing results consistent with the boundary energies calculated by Wolf [22][23][24] and with experimental boundary energies measured by the thermal grooving method [18]. ...
... Wolf [22][23][24] showed that the energy anisotropies of Fe and Mo were similar for symmetrical tilt boundaries, twist boundaries on (1 0 0) and (1 1 0) planes, and certain general grain boundaries. Morita and Nakashima [20] investigated the boundary energy of <0 0 1> symmetric tilt boundaries in Mo, producing results consistent with the boundary energies calculated by Wolf [22][23][24] and with experimental boundary energies measured by the thermal grooving method [18]. Tschopp et al. [21] examined a large data set of grain boundary energies in Fe using molecular statics. ...
Atomistic simulations using the embedded atom method were employed to compute the energies of 408 distinct grain boundaries in bcc Fe and Mo. This set includes grain boundaries that have tilt, twist, and mixed character and coincidence site lattices ranging from Σ3 to Σ323. The results show that grain boundary energies in Fe and Mo are influenced more by the grain boundary plane orientation than by the lattice misorientation or lattice coincidence. Furthermore, grain boundaries with (1 1 0) planes on both sides of the boundary have low energies, regardless of the misorientation angle or geometric character. Grain boundaries of the same type in Fe and Mo have strongly correlated energies that scale with the ratio of the cohesive energies of the two metals.
... For two-dimensional symmetric tilt GBs, the 13 boundaries, whose misorientation angles are near θ = 30 • , show an energy cusp. Generally, in the GBs in the three-dimensional models, the large-angle boundary exhibits the maximum GB energy, and energy cusps can be observed at some GB misorientation angles depending on the simple combination of the structural units [22,41,[54][55][56][57][58][59]. Therefore, even the GBs with two-dimensional models here can capture the characteristics of general GBs, enabling the qualitative analysis of the dependence of the GB structure on the dislocation emission from the GBs. ...
... From the inset of Fig. 3, we can confirm a strong correlation between the GB energy and the GB free volume. The positive correlation has been previously reported from three-dimensional MD calculations [22,[55][56][57][58][59][63][64][65] and first-principles calculations [23,66]. Therefore, similar to the GB free volume, the GB energy also decreases with an increase in δ or a decrease in T a . ...
As one of the promising candidates for developing next-generation structural materials, high-entropy alloys (HEAs) have recently attracted significant interest because of their unique mechanical properties, including their coexisting of high strength and ductility properties. Here, through atomic simulations, we demonstrate that the segregation of elements to grain boundaries (GBs) due to atomic-size differences, which is one of the most important characteristics of HEAs, contributes to the coexistence of high strength and high ductility in HEAs. To focus on only the effect of the size difference on the GB segregation, ignoring the difference in the chemical bonding energies among all the constituent elements, we employ two-dimensional virtual quinary HEA models. The HEAs are subjected to tensile and compressive load tests, and the stress required for dislocation emission from the GBs is measured. We demonstrate that the GB segregation in the HEAs increases the stress required for dislocation emission from the GBs, thereby increasing the strength of the HEAs. This is because the GB segregation in the HEAs stabilizes the GB structure by decreasing the GB free volume. Notably, the GB segregation also decreases the heterogeneity of the mechanical field between the grain interiors and the GBs, which is an intrinsic attribute of ordinary materials, and the homogenization of the mechanical field can improve the ductility of HEAs, preventing intergranular fracture. Our results can serve as a guide for designing HEAs with both high strength and high ductility through the effective utilization of GB segregation.
... Meanwhile, there are various modelling methods to construct grain boundary structures. Some previous articles [37][38][39] constructed grain boundaries by matching two free surfaces with different Miller indexes or rotating crystal parts according to CSL parameters [40]. Then, the stable grain boundary structures were obtained by energy minimization methods [39,41]. ...
... The STGBs could also be constructed in a non-periodic system [42,43], allowing STGB simulations with small system size and without pressure control. However, the grain boundaries established by these defined structures [37][38][39][40][41] are always ideal, which may ignore the temperature, pressure and size effects in the energy minimization process. With the developments of computer science, the large-scale molecular dynamics simulation enables us to access the natural formation process of grain boundaries [44,45]. ...
A high efficient hybrid Monte Carlo and molecular dynamics (MC/MD) approach was carried out to reproduce silicon [001] small angle symmetric tilt grain boundaries at the atomic level. In the present paper, we concentrated on the variations of grain boundary properties under various misorientation angles, from which three critical misorientation angles were proposed and four grain boundary stages were divided to illustrate the structural transition of grain boundaries from small to large angle and the corresponding characteristics. The roles of dislocation stress fields in shaping some special grain boundary properties, such as dislocation structures and defect sink, were discussed. We also calculated three critical parameters (dislocation core radius, dislocation core energy and grain boundary width) about structures and energies of these studied grain boundaries, which would benefit the subsequent grain boundary engineering. Further investigations indicated that the three critical misorientation angles could be converted to the average distance of adjacent dislocations, which equalled to 6-, 2- and 1-times of dislocation core radius. Compared with the published literatures and classical theories, the simulations showed well accordance in structural units, stress fields and elastic strain energies.
... A moderate correlation between GB energy and GB excess free volume was found. Wolf [56,57,58] investigated the correlation between the GB energy and the excess free volume for both FCC and BCC metals via atomistic simulations. Their finding suggested that GB energy and excess free volume are strictly related. ...
... Their finding suggested that GB energy and excess free volume are strictly related. Wolf [56,57,58] also noticed that the strength of such correlation (and therefore the scatter in the data) is strictly related to the choice of the MD potential. Cao et al. [5] argued that the moderate correlation between the GB energy and excess free volume found by Olmsted et al. [37,38] can be improved by using by density functional theory results. ...
This report describes three improvements made to a physically-based model for creep and creep rupture in Grade 91 steel developed as part of the Advanced Reactor Technologies program:
1. Progress on transitioning the model framework to the MOOSE finite element package to improve parallel scalability, increase the physical size of the simulations, and incorporate multiphysics effects into the model.
2. The development of a physically-based method of scaling the model parameters to accurately capture creep in Grade 91 for temperatures in the range of 450 to 500C.
3. Extending the model to account for microstructural statistical variations, specifically capturing the effect of grain boundary energy on the physical parameters underling the model for grain boundary void nucleation and cavitation.
Put together, these improvements result in an accurate, predictive model for the physical response of Grade 91 steel over the expected use temperature range for the material in future liquid metal cooled fast reactors. The model can be used to more accurately predict engineering properties of the material for very long service lives, which could lead to safer, more economical future advanced reactors.
... These studies have included both symmetric and asymmetric tilt boundaries on (111), (100), (110) and (113) planes [14,18]. In contrast, only a handful of studies exist on bcc transition metals including the work of Wolf on Fe/Mo [12,19] [21]. There are few experimental and/or simulation reference points for atomistic GB structures in tantalum; the principal reference is the structure of the Σ5 (310)/[001] CSL tilt boundary investigated both experimentally and theoretically by Campbell et al. [22,23]. ...
... GBEs for Ta are shown as black asterisks (GBE values can be found in the supplemental material). Also provided is relevant data for bcc Fe (blue) and bcc Mo (red) from empirical potentials (Wolf[12,19], Morita and Nakashima[31], and Tschopp et al.[6]) illustrating similar trends. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) ...
Grain boundaries can play a significant role in the mechanical response of materials. Atomistic simulations are used to investigate 79 coincidence site lattice grain boundary structures and energies in tantalum, a model body-centered cubic transition metal. Quasi-symmetric Σ3, Σ5, Σ7, Σ13, and Σ27 boundaries are observed, of which Σ3 and Σ7 also exist as traditional mirror-symmetry conserving boundary structures. These results are supported by previous observations of similar phenomena in other bcc transition metal Σ5 boundaries. Metastable low energy Σ3 boundary structures in tantalum could influence the formation and stability of deformation twins and abnormal growth grain favoring Σ3 boundaries.
... Multiple wide-ranging atomistic studies of GB energy using IPs have been conducted. One of the first broad surveys was performed by Wolf [22][23][24][25][26][27], who considered symmetric and asymmetric tilt and twist boundaries in face-centered cubic (fcc) and body-centered cubic (bcc) materials. Still, exploration of GB energy for various materials, IPs, and crystal structures is a topic of ongoing research [28]. ...
We present a systematic methodology, built within the Open Knowledgebase of Interatomic Models (OpenKIM) framework (https://openkim.org), for quantifying properties of grain boundaries (GBs) for arbitrary interatomic potentials (IPs), GB character, and lattice structure and species. The framework currently generates results for symmetric tilt GBs in cubic materials, but can be readily extended to other types of boundaries. In this paper, GB energy data are presented that were generated automatically for Al, Ni, Cu, Fe, and Mo with 225 IPs; the system is installed on openkim.org and will continue to generate results for all new IPs uploaded to OpenKIM. The results from the atomistic calculations are compared to the lattice matching model, which is a semi-analytic geometric model for approximating GB energy. It is determined that the energy predicted by all IPs (that are stable for the given boundary type) correlate closely with the energy from the model, up to a multiplicative factor. It thus is concluded that the qualitative form of the GB energy versus tilt angle is dominated more by geometry than the choice of IP, but that the IP can strongly affect the energy level. The spread in GB energy predictions across the ensemble of IPs in OpenKIM provides a measure of uncertainty for GB energy predictions by classical IPs.
... The results of such studies may be relevant to the Li case as these metals are non-ferromagnetic in contrast to iron (particularly, comparative energy of interstitial atom configurations in iron was predicted to differ from the other bcc metals [23]). There are numerous theoretical works aimed at finding the correlation between the GB energy and its structure [27][28][29][30][31][32][33] described by the macroscopic degrees of freedom (i.e., the rotation axis, the misorientation angle, and the GB plane) and the microscopic configuration (obtained by shifting the crystallites and/or removing the overlapping atoms [29]). However, most of these studies focus on covering as many GB structures as possible, and, therefore, they were restricted to static calculations at 0K. ...
Whisker growth during lithium electrodeposition is one of the most critical factors limiting the safe operation of high-capacity lithium-metal batteries. According to the recent findings, the underlying fundamental mechanism includes solid-state mass transport occurring presumably along the grain boundaries (GBs) and feeding the whiskers. Here, we perform atomistic simulations of lithium for both (100) and (110) twist GBs using a high-accuracy machine learning interatomic potential. We find that at room temperature the GBs under study are liquified and possess a high self-diffusion coefficient comparable to that of the liquid phase. The grain growth via absorption of additionally deposited atoms at the GBs to build a new atomic layer of the crystal lattice (i.e., layer-by-layer growth) is explicitly modeled, which enables uncovering various atomistic configurations of the grain boundaries that may take place in the course of Li electrodeposition. Additionally, in order to assist further atomistic simulations of polycrystalline Li, we analyzed the capability of the existing conventional EAM-based force fields to model the GB behavior.
... In these ORs, 100% of pairs of the two main variants made a P 3 boundary. It should be emphasized that the boundary energy of CSLs is much lower than other high angle boundaries, especially in a P 3 boundary [64,66,67]. The diffusional process of self-organization would generate a high fraction of GBs in the initial nanoligaments, and the selected variants preferentially transform under the strict rule of arranging CSLs to reduce the internal energy of the nanostructure. ...
Liquid metal dealloying (LMD) is a promising technique that can be used to synthesize non-noble porous materials by preventing oxidation through using a metallic melt. However, the phase transformation behavior between a parent grain and synthesized ligaments surrounded by penetrating liquid metal channels remains unknown, despite its importance on the final physical properties. In this study, the temperature effect on the transformation mechanisms during the LMD process is investigated. At a low temperature of 600 °C, a fcc grain of (FeCo)xNi100−x precursors transforms to bcc FeCo ligaments by following unique orientation relationships (ORs), which differ from the well-known ORs like Bain, Kurdjumov-Sachs and Nishiyama-Wassermann. A few select variants are generated in the proposed ORs, and the formation of coincident site lattice boundaries is shown to play a crucial role in the variant selection to reduce the internal energy of the initial nanostructure. However, at a higher temperature of 800 °C, the crystal orientation of ligaments prefers to align in the direction, which is parallel to the flow direction of Mg melt, owing to a strong interaction between the melt and ligaments. Ligaments are elongated along with the fiber texture, and this transformation behavior is irrelevant to the parent orientation. Keywords: Liquid metal dealloying, Phase transformation, Orientation relationship, Coincident site lattice, Interface structure, Liquid-solid reactions
... Previous computational investigations have been performed investigating grain boundary energies in bcc Mo by Morita [16] and Wolf [17,18] utilizing a Finnis-Sinclair potential [19]. Yesilleten and Arias [20] utilized a model generalized pseudopotential theory (MGPT) [21] potential to investigate bcc Mo grain boundaries and their interactions with vacancies. ...
A monolithic fuel design based on a U-Mo alloy has been selected as the fuel type for conversion of the United States High-Performance Research Reactors (HPRRs). A 2015 post-irradiation examination (PIE) report showed accelerated swelling in U-10Mo fuels at fission densities much lower than previously observed. This PIE report showed a large amount of compositional banding, or regions of low Mo content adjacent to regions of high Mo content, with low Mo content typically along grain boundaries. Lower Mo content can lead to phase decomposition from the gamma U-Mo body-centered cubic phase to the alpha U phase as well as an earlier onset of recrystallization. Thus, the phenomenon of Mo depletion at grain boundaries is an important factor in the accelerated swelling behavior of U-Mo fuel. However, the physical origin of Mo depletion at grain boundaries is still unclear. In this work, molecular dynamics simulations have been performed to calculate the grain boundary and surface energies of body-centered cubic (bcc) U, bcc Mo and alloys of U-Mo from 600 K to 1200 K. It is observed that the lower grain boundary energy of bcc U, compared to bcc Mo, provides the driving force for Mo depletion at grain boundaries. This driving force diminishes with increasing temperature, but is not eliminated. This information can be utilized as inputs to higher length scale modeling methodologies and provide specification guidance to fabricators.
... In the common modeling approach, also known as the γ-surface method, a GB is constructed by joining two perfect half-crystals with different orientations while sampling the possible translations of the grains relative to each other. This methodology has been employed to predict structures and energies of GBs including those in bcc materials [36,37,38,39,40,41,42,43]. The γ-surface approach has been challenged by a number of computational studies of GBs in several different materials systems. ...
We use atomistic simulations to investigate grain boundary (GB) phase transitions in el- emental body-centered cubic (bcc) metal tungsten. Motivated by recent modeling study of grain boundary phase transitions in [100] symmetric tilt boundaries in face-centered cu- bic (fcc) copper, we perform a systematic investigation of [100] and [110] symmetric tilt high-angle and low-angle boundaries in bcc tungsten. The structures of these boundaries have been investigated previously by atomistic simulations in several different bcc metals including tungsten using the the {\gamma}-surface method, which has limitations. In this work we use a recently developed computational tool based on the USPEX structure prediction code to perform an evolutionary grand canonical search of GB structure at 0 K. For high-angle [100] tilt boundaries the ground states generated by the evolutionary algorithm agree with the predictions of the {\gamma}-surface method. For the [110] tilt boundaries, the search predicts novel high-density low-energy grain boundary structures and multiple grain boundary phases within the entire misorientation range. Molecular dynamics simulation demonstrate that the new structures are more stable at high temperature. We observe first-order grain boundary phase transitions and investigate how the structural multiplicity affects the mechanisms of the point defect absorption. Specifically, we demonstrate a two-step nucleation process, when initially the point defects are absorbed through a formation of a metastable GB structure with higher density, followed by a transformation of this structure into a GB interstitial loop or a different GB phase.
... Symmetric tilt grain boundaries have equal but opposite tilt angle of two grains about a common tilt axis. The variation of grain boundary energy with tilt angle depends on the tilt axis the material [2,3]. ...
... Figure 11 demonstrates that the crystal responds to the applied strain by twinning. The highlighted {111} fcc planes have an hcp-like coordination; they mark coherent Σ3 twin boundaries and separate twin and matrix grains [Finnis and Rúhle, 1993;Wolf, 1989]. The larger volumes in Fig. 11 still retained the original orientation while the material enclosed between the narrow pairs of twin boundaries has re-oriented to accommodate the large stress developing. ...
Using molecular-dynamics simulation, we study the phase transformations in Fe thin films induced by uni- and biaxial strain. Both the austenitic transformation of a body-centered cubic (bcc) film at the equilibrium temperature of the face-centered cubic (fcc)–bcc transformation and the martensitic transformation of an undercooled fcc film are studied. We demonstrate that different strain states (uni- or biaxial) induce different nucleation kinetics of the new phase and hence different microstructures evolve. For the case of the austenitic transformation, the direction of the applied strain selects the orientation of the nucleated grains of the new phase; the application of biaxial strain leads to a symmetric twinned structure. For the martensitic transformation, the influence of the strain state is even more pronounced, in that it can either inhibit the transformation, induce the homogeneous nucleation of a fine-dispersed array of the new phase resulting in a single-crystalline final state, or lead to the more conventional mechanism of heterogeneous nucleation of grains at the free surfaces, which grow and result in a poly-crystalline microstructure of the transformed material.
... For the ( )( ) 114 011 ATwGB we see a marked difference between the behavior of the atomistic data for copper vs for gold, indicating that for this type of interface Fig. 8. BCC STwGB results of the covariance model (relaxed and unrelaxed) for α-iron, compared with atomistic results using a Johnson potential. MD data from Wolf (1989aWolf ( , 1991. geometry may not be the main factor influencing GB energy. ...
The energy density of crystal interfaces exhibits a characteristic ‘cusp’ structure that renders it non-convex. Furthermore, crystal interfaces are often observed to be faceted, i. e., to be composed of flat facets in alternating directions. In this work, we forge a connection between these two observations by positing that the faceted morphology of crystal interfaces results from energy minimization. Specifically, we posit that the lack of convexity of the interfacial energy density drives the development of finely faceted microstructures and accounts for their geometry and morphology. We formulate the problem as a generalized minimal surface problem couched in a geometric measure-theoretical framework. We then show that the effective, or relaxed, interfacial energy density, with all possible interfacial morphologies accounted for, corresponds to the convexification of the bare or unrelaxed interfacial energy density, and that the requisite convexification can be attained by means of a faceting construction. We validate the approach by means of comparisons with experiment and molecular dynamics simulations including symmetric and asymmetric tilt boundaries in face-centered cubic (FCC) and body-centered cubic (BCC) crystals. By comparison with simulated and experimental data, we show that this simple model interfacial energy combined with a general microstructure construction based on convexification is able to replicate complex interfacial morphologies, including thermally-induced morphological transitions.
... A calculation of the average boundary energy along these three low-index misorientation axes (allowing for a tolerance of 10º) confirms this impression and shows that the relative boundary energies relate as E <100> : E <111> : E <110> = 109 : 106 : 99. Similar conclusions obtained by molecular dynamics simulations were published by Wolf [19] taking into consideration symmetrical tilt boundaries only. ...
It is often assumed that the texture formation during solid state transformations in low carbon steels critically depends on the local crystallographic misorientation at the interface between transformed and not yet transformed material volume. In some cases, a theoretical crystallographic orientation relation can be presumed as a necessary prerequisite for the transformation to occur. Classical examples of such misorientation conditions in steel metallurgy are the orientation relations between parent and product grains of the allotropic phase transformation from austenite to ferrite (or martensite) or the hypothetical 〈110〉26.5° misorientation between growing nuclei and disappearing grains in a recrystallization process. One way to verify the validity of such misorientation conditions is to carry out an experiment in which the transformation is partially completed and then observe locally, at the transformation interface, whether or not the presumed crystallographic condition is complied with. Such an experiment will produce a large set of misorientation data. As each observed misorientation Δg is represented by a single point in the Rodrigues-Frank (RF) space, a distribution of discrete misorientation points is obtained. This distribution is compared with the reference misorientation Δgr, corresponding to a specific physical condition, by determining the number fraction dn of misorientations that are confined within a narrow misorientation volume element dω around the given reference misorientation Δgr. In order to evaluate whether or not the proposed misorientation condition is obeyed, the number fraction dn of the experimentally measured distribution must be compared with the number fractions dr obtained for a random misorientation distribution. The ratio dn/dr can be interpreted as the number intensity f i of the given reference misorientation Δgr. This method was applied on the observed local misorientations between the recrystallizing grains growing into the single crystal matrix of a Fe-2.8%Si alloy. It was found that the number intensity of the 〈110〉26.5° misorientation increased with a factor 10 when the misorientation distribution was evaluated before and after the growth stage. In another example the method was applied to the misorientations measured at the local interface between parent austenite and product martensite grains of a partially transformed Fe-28%Ni alloy. It could be established that the Nishiyama-Wasserman relations ({111}γ//{110}α 〈112〉γ//〈110〉α) prevail over the Kurdjumov-Sachs relations ({111}γ//{110}α and 〈110〉γ//〈111〉α) although a considerable scatter was observed around either of the theoretical correspondences. A full parametric misorientation description was also applied to evaluate the relative grain boundary energies associated with a set of crystallographic misorientations observed near triple junctions in Fe-2%Si. In this instance it was found that the boundaries carrying a misorientation of the type 〈110〉ω carry a lower interfacial energy than the 〈100〉 or 〈111〉 type boundaries.
Niobium is an important element in Zr–Nb alloy, which is widely employed for structural material in many nuclear reactors. Superior mechanical properties and creep resistance in conjunction with low neutron cross-section, makes Zr–Nb alloy a suitable candidate for the manufacturing of pressure tubes and fuel cladding in nuclear reactors. Molecular dynamics based simulations were performed to study the point defect dynamics in niobium. Atomistic simulations were performed in the environment of classical mechanics to investigate the effect of grain boundary configurations on the point defect dynamics. Symmetric and asymmetric tilt grain boundaries were generated between the two crystals of niobium, and defect formation and migration energies were calculated as a function of distance from the grain boundary plane. A relationship between the grain boundary energies and point defect formation energies was predicted, and it was concluded that higher energy grain boundaries have higher tendency to behave as sink for point defects. Lower offset in vacancy formation energies between the initial and terminating position of vacancy, energetically favours the migration of point defects, whereas higher energy offset between bulk and grain boundaries require a more energy to migrate vacancy towards the grain boundary.
A structural vacancy model of tilt grain boundaries in metals has been developed. For construction of the stable structure of the boundary, the initial pattern was chosen according to the CSL model. The introduction of additional atoms and vacancies into the boundary region and shifting of atoms by the interatomic forces stabilize its structure. The criterion of a stable structure is the grain-boundary energy. The comparison of two main approaches to the stabilization of the grain structure demonstrated that changing the number of atoms at the boundary is more energetically advantageous than the relative shift of grains. The stability of the structure obtained has been studied under the shear stress. In the model developed, atomic structures obtained with pair and many-body potentials have been compared. The comparative analysis has shown that the grain-boundary structure does not depend on the choice of potential; atomic positions differ by less than 0.1 Å, which is 2.5% of the lattice parameter. The atomic structure is in agreement with experimental images of grain boundaries.
In this study, the grain boundary energy of symmetric and asymmetric iron bicrystals is calculated for Σ3, Σ9 boundaries with 110 tilt axis and Σ5 with <100> tilt axis. The calculations are carried out using molecular dynamics simulations with the embedded-atom method potential. A modified method for creating grain boundary atomic structure is proposed that has sufficient accuracy and its computation cost will be considerably lower than the previously used methods. The effect of three parameters namely rigid body movement, overlapping distance, and reduction side is investigated and compared to previous studies and the optimal parameters are introduced which leads to a better performance in bicrystal modeling.
Uranium-silicide (U-Si) fuels are being pursued as a possible accident tolerant fuel (ATF). This uranium alloy benefits from higher thermal conductivity and higher fissile density compared to uranium dioxide (UO2). In order to perform engineering scale nuclear fuel performance simulations, the material properties of the fuel must be known. Currently, the experimental data available for U-Si fuels is rather limited. Thus, multi-scale modeling efforts are underway to address this gap in knowledge. Interfaces play a critical role in the microstructural evolution of nuclear fuel under irradiation, acting both as sinks for point defects and as preferential nucleation sites for fission gas bubbles. In this study, a semi-empirical modified Embedded-Atom Method (MEAM) potential is utilized to investigate grain boundaries and free surfaces in U3Si2. The interfacial energy as a function of temperature is investigated for ten symmetric tilt grain boundaries, eight unique free surfaces and voids of radius up to 35 Å. The point defect segregation energy for both U and Si interstitials and vacancies is also determined for two grain boundary orientations. Finally, the entropy change and free energy change for grain boundaries is calculated as a function of temperature. This is the first study into grain boundary properties of U-Si nuclear fuel.
We predict structure and energy of low-angle (11¯0) pure twist grain boundaries (GBs) in five BCC transition metals (β-titanium, molybdenum, niobium, tungsten, and tantalum) using a combination of atomistic and microscopic phase-field (MPF) modeling. The MPF model takes as inputs solely the generalized stacking fault energy surfaces (i.e., the γ-surface) and elastic constants obtained from the atomistic simulations. Being an energy-based method, the MPF model lifts the degeneracy of the geometric models in predicting GB structures. For example, the multiple indefinite solutions offered by the Frank-Bilby equation are shown to converge to exactly the same equilibrium structure. It predicts a transition of the equilibrium GB structure from a pure screw hexagonal network (Mo and W) to mixed hexagonal networks (Nb and Ta) to a rhombus network (β-Ti) of dislocations. Parametric simulation studies and detailed analyses of the underlying dislocation reactions that are responsible for the formation of the rhombus and hexagonal structures reveal a close correlation between material properties (including the elastic anisotropic ratio and the local curvature on the γ-surface) and the GB structure and energy in BCC metals. This integrated approach allows one to explore, through high throughput calculations, the potential to tailor the structure and energy of special GBs in BCC metals by alloying.
We report a computational discovery of novel grain boundary structures and multiple grain boundary phases in elemental bcc tungsten. While grain boundary structures created by the \gamma-surface method as a union of two perfect half crystals have been studied extensively, it is known that the method has limitations and does not always predict the correct ground states. Here, we use a newly developed computational tool, based on evolutionary algorithms, to perform a grand-canonical search of a high-angle symmetric tilt boundary in tungsten, and we find new ground states and multiple phases that cannot be described using the conventional structural unit model. We use MD simulations to demonstrate that the new structures can coexist at finite temperature in a closed system, confirming these are examples of different GB phases. The new ground state is confirmed by first-principles calculations.
Energy and structure of tilt grain boundaries are studied in Nickel-Titanium (NiTi) alloys. Molecular dynamics (MD) simulations are utilized to investigate the excess energy of symmetric and asymmetric tilt grain boundaries as a function of the misorientation and inclination angles. Structural units of different symmetric grain boundaries are identified and the correlation between the structure and energy is investigated. It is shown that the Read-Shokley model can accurately predict the energy of low angle symmetric tilt grain boundaries in austenite NiTi alloy. Density functional theory (DFT) calculations are used to study two representative symmetric grain boundaries. Comparing to DFT calculations, it is shown that the MD results can predict the GB potential energies accurately. The Electron charge density distributions are studied and the bonding strength between atoms, and low/high charge density regions are investigated, and accosted with the structural units in the grain boundaries. It is shown that a symmetric arrangement of Ni and Ti atoms at the GB improved the uniformity of bonding and charge density distribution.
Atomistic modeling was used to investigate the energetics and structure of the Bagaryatskii orientation relationship between ferrite and cementite within pearlite. The atomic level results show that the interface structure consists of a rectangular array of dislocations that lie along the high symmetry directions of the interface. The interface can be constructed using three different atomic terminating planes in the cementite structure, which dictates the chemistry and registry of the interface and controls the interfacial energy. The FeC-Fe terminating plane is always the lowest energy because the interfacial dislocations are most easily able to spread on these planes, thus reducing the interfacial energy. These atomistic results compare favorably with results from a continuum model based on O-lattice theory and anisotropic continuum theory. The dislocation spacing and interfacial energies predicted from the continuum level agree well with the atomistic simulation results.
Grain-boundary (GB) structure and properties are usually analyzed in terms of ground-state (minimum-energy) GB states. However, global equilibrium is rarely achieved in materials. In this paper, we investigate the nature of GB metastability and its impact on material properties. Higher-energy GB states can be the result of nonequilibrium processes or simply thermal excitations. While the existence of limited GB metastability is widely known for a few simple GBs, we demonstrate that the multiplicity of metastable GB states is, in general, very large. This conclusion is based upon extensive atomistic bicrystal simulations for both symmetric tilt GBs and twist GBs in three very different materials. The energies of these GB states are densely distributed so that the dependence of the GB energy on misorientation is better described as an energy band rather than as a single curve as in the traditional picture. Based upon the distribution of metastable GB states, we introduce a GB statistical-mechanics picture and apply it to predict finite-temperature equilibrium and nonequilibrium properties. When GB multiplicity exists, GB structures can be thought of as domains of different GB states separated by various classes of line defects. The existence of a large set of metastable GB states, very close in energy, suggests an analogy between the behaviors of GBs and glasses and implies the potential for GB engineering.
In the present study we investigate the microstructure of tempered martensite ferritic steels. It is well known that inside former austenite grains and inside packets of former martensite laths ultrafine micro grains (average size near: 1 μm) govern the strength of this material class. Micro grain boundaries are decorated by carbides (average size after creep near: 0.05 μm). However, in transmission electron micrographs it is commonly found that there are micro grain boundaries with a high carbide density while there are others where no carbides can be detected. In the present study we make an attempt to decide whether the crystal log raphic character of micro grain boundaries can be related to the number density of carbides at the boundaries. Kikuchi line diffraction patterns were used to determine the misorientation angle between two adjacent micro grains; we select only micro grain boundaries which represent 〈110〉 - and 〈100〉 - twist boundaries. A quantitative microstructural analysis was performed to determine the density of carbides on boundaries. Our results are discussed on the basis of general tendencies which were reported for grain boundaries in the literature.
Metal and complex hydrides offer very promising prospects for hydrogen storage that reach the DOE targets for 2015. However, slow sorption kinetics and high release temperature must be addressed to make automotive applications feasible. Reducing the enthalpy of formation by destabilizing the hydride reduces the heat released during the hydrogenation phase and conversely allows desorption at a lower temperature. High-energy ball milling has been shown to decrease the release temperature, increase reaction kinetics and lower the enthalpy of formation in certain cases. Increased surface and grain boundary energy could play a role in reducing the enthalpy of formation, but the predicted magnitude is too small to account for experimental observations. As the particle and grain sizes are reduced considerably under high-energy treatments, structural defects and deformations are introduced. These regions can be characterized by an excess volume due to deformations in the lattice structure, and have a significant effect on the material properties of the hydride. We propose a thermodynamic model that characterizes the excess energy present in the deformed regions to explain the change in physical properties of metal hydrides. An experimental investigation using the TEM to study the effect of lattice deformations and other nanostructures on the desorption process is underway.
The molecular-dynamic method has been used to study the interaction of lattice vacancies with symmetrical grain boundaries (GBs) in aluminum. The fraction of trapped vacancies has been found to depend linearly on the distance to the GB plane. The average velocity of the vacancy migration toward the boundary decreases exponentially with an increase in the distance between the GB plane and vacancy. The radius of the region of trapping of a vacancy by the boundary is limited to two to three lattice parameters and grows with an increase in temperature. Four types of boundaries, which are characterized by different capability for the trapping of vacancies, have been determined.
The transformation of an amorphous Ni-P alloy was investigated calorimetrically. In particular, the formation of the interfaces in nanocrystalline Ni-P samples with grain sizes ranging from a few nanometers to 60 nm was studied. A linear increase in the interracial excess energy was found with increasing grain size on a nanometer scale. However, a rather small interracial excess energy was obtained when the grain size was a few nanometers. The variation of interracial excess energy with grain size was found to correlate with that of the interracial excess volume derived from experimental results.
Diffusion of Ag in a nanometer Ni‐B‐O amorphous alloy has been studied by heavy‐ion Rutherford backscattering spectroscopy. A significant mass transport was observed at the extremely low anneal temperatures. The values of the diffusion coefficients that were found at 293, 323, and 373 K are 3.5×10−16, 4.3×10−15, and 5.0×10−14 cm2/s, respectively. Moreover, small values for the activation energy, 0.58 eV, and the pre‐exponential factor of the diffusion coefficient, 3.9×10−6 cm2/s, were derived from the Arrhenius plot. Finally, a possible free volume controlled diffusion mechanism was discussed.
The effect of boundary faceting on densification has been studied in a 5mol.% TiO2–excess BaTiO3 model system. It was possible to inhibit grain growth during sintering in moderately reducing atmospheres after a hydrogen presintering treatment and, thus, to exclude the effect of grain growth on densification. For a given oxygen partial pressure (PO2) there was a saturated relative density which decreased with increasing PO2 and boundary faceting. The fraction of small pores was also higher in the samples sintered under high PO2 than in those sintered under low PO2. These experimental observations demonstrate the presence of a critical driving force for densification in systems with faceted boundaries and its dependence on boundary faceting, i.e. the critical driving force is larger for systems with greater faceting. The result further confirms that a faceted boundary, either fully or partially faceted, is an imperfect atom source for densification; this idea is in contrast to the conventional understanding.
The recrystallization behaviours induced by three different cold rolling and annealing processes are analyzed. These are shown to be compatible with three distinct types of nucleation phenomena based on: i) random, ii) low stored energy, and iii) high stored energy nucleation. Growth is considered to follow nucleation sequentially and to involve more rapid growth into neighbouring grains that are related to the nucleus by means of Σ 〈110〉 rotations. Instead of individual CSL relationships, the concept of {110} plane matching between the nucleus and matrix grain being consumed is employed. This is justified in terms of the very low free volumes associated with the boundaries between Σ 〈110〉 related grains; the low capacity of these boundaries for the absorption of solute atoms is considered to be responsible for their high mobility. In heavily rolled materials in which dislocation sheets lie close to the maximum shear stress {110} planes, only the planes containing sheets appear to participate in the plane matching process. Recrystallization textures are simulated using the nucleation and growth model described above; these are in good agreement with the experimental observations in the case of the present three steel processing operations.
Experimental IF steel textures produced by cold rolling and annealing are analyzed in terms of a sequential nucleation and growth model. High stored energy (high Taylor factor) nucleation is assumed to take place in the 85% cold reduced material. Growth of these nuclei is then considered to occur by means of Sigma < 110 > CSL transformations applied to the discretized orientation distribution. A type of variant selection is employed, according to which only the < 110 > axis closest to one of the maximum shear stress poles is permitted to operate. The individual effects of the occurrence of Sigma 9, Sigma 11, Sigma 17c, Sigma 19a, Sigma 33a and Sigma 33c transformations is demonstrated. It is concluded that the experimental results are well represented by a combination of Sigma 17c, Sigma 19a acid Sigma 33a growth, with perhaps an additional contribution from Sigma 9.
Differences and similarities of structure-property relationships observed in nanostructured materials prepared by electrodeposition and consolidation of nanocrystalline precursor powders will be discussed. Directionally similar properties observed in both classes of materials, such as initial hardness, strength and electrical resistivity, can be explained in terms of enhanced volume fractions of grain boundaries and triple junctions which are common to both types of structures. However, for properties that show considerable differences such as saturation magnetization, thermal expansion, heat capacity or Young's modulus, other microstructural characteristics must be considered.
Nanocrystalline (NC) NiP materials with average grain sizes ranging from 7 to 48 nm were synthesized by crystallization of amorphous alloys. The thermal instabilities of these samples, which are characterized by the nanometer-sized grain growth, were studied by using a differential scanning calorimeter (DSC). Experimental evidences indicate that with a decrease of the grain size, the starting temperature, the peak temperature, and the activation energy for the grain growth process increase significantly. The tendency of a decreasing instability in the smaller grain-sized sample is contradictory to the classical theory of grain growth in conventional polycrystalline materials. The anomalous instability was analyzed following an experimental fact that the excess energy and the excess volume of the interfaces in the NC materials are decreased with a reduction of the grain size.
This paper provides a computer modelling technique, associated with the molecular statics relaxation method, for the study of nanostructured crystals. Atomistic simulations have been carried out to study the structural features of nanocrystals, such as the reduced mass density, the boundary component proportion, the excess lattice parameter, the excess atomic volume and the radial distribution function. The energy and elastic properties also have been investigated. The simulated results have shown considerable agreement with the experimental results, which implies that the present modelling method is suitable to be used to study the structures and properties of nanocrystals.
Thermal expansion behaviors of nanocrystalline (NC) Ni-P alloy samples with grain sizes ranging from a few to 127 nm were studied experimentally using thermal mechanical analysis. Porosity-free NC Ni-P samples with a wide grain size range were synthesized by completely crystallizing amorphous Ni-P alloy at different annealing temperatures. Measurements showed that the linear thermal expansion coefficient (αL) increases markedly with a reduction of the average grain size in the as-crystallized NC Ni-P samples. The thermal expansion coefficient was also found to decrease during grain growth in the as-crystallized NC sample upon annealing. From the grain size dependence of αL in these NC samples, we deduced that the difference in thermal expansion coefficients between the interfaces and the nm-sized crystallites diminishes when the grain size becomes smaller. This tendency agrees well with other experimental results on the structural characteristics of the interfaces and the nm-sized crystallites in NC materials.
A thermodynamics for the phase transformation from γ(fcc) to α(bcc) in nanocrystalline (NC) Fe is considered. Gibbs free energies of the interfaces in NC γ- and α-Fe particles were calculated, respectively, by means of a quasiharmonic Debye approximation, yielding a larger increase in the total Gibbs free energy of α-Fe than that of γ-Fe. This is attributed to the difference in their interfacial energies. As a result, the fcc NC Fe can be thermodynamically stable at room temperature when the grain size is sufficiently small. Taking into account the thermodynamic equilibrium condition, the critical grain size for the γ-Fe phase to exist in stable form at 300 K was quantitatively calculated for different excess volumes ΔV, a parameter describing the state of interface based on a dilated crystal model. The assumptions made in the present model and the factors influencing the critical grain size are discussed.
The development of textures in interstitial-free (IF) steels as a result of annealing after cold rolling is described with
the help of a combined nucleation and growth model. Nucleation is simulated by assuming that high stored energy nucleation
occurs preferentially in high Taylor factor regions in the 75 to 85 pct cold reduced materials. Growth of the nuclei then
takes place by means of Σ (110) type as well as by Σ 7 (111) type coincident site lattice (CSL) transformations. Of the six
symmetrically equivalent (110) transformation axes, only the ones near the maximum shear stress poles are assumed to operate.
The effects of the migration of individual Σ 9, Σ 11, Σ 17c, Σ 19a, Σ 33a, and Σ 33c (110) boundaries are analyzed. Their
relative mobilities and contributions to the final texture are deduced by matching the simulated and experimental preferred
orientations using a /ldleast-squares” method. On the basis of experimental results for two steels, the various boundary types
are observed to have the following mobility ratios: Σ 33a: 12, Σ 19a:4, Σ 9:1, Σ 33c:l, and Σ 17c: 2.
An atomistic study of tilt grain boundary structures in f.c.c. metals has been made. The principal aim of this study is to understand the structure of long-period (`general') tilt boundaries. Boundaries for which Sigma
It is suggested that the "supermodulus effect" observed for composition-modulated superlattices arises from the presence of the structurally disordered solid interfaces and not necessarily from electronic structure effects.
Un nombre de structures non-équivalentes ont été trouvées dans les études atomistiques des joints de grains périodiques. Ces structures peuvent se transformer mutuellement en ôtant, ou en ajoutant des lamelles d'atomes qui sont parallèles aux joints [19]. Cette multiplicité résultante des structures est très extensive pour des joints de grains généraux avec de longues périodes. On montre ici que les configurations de lacune d'énergie plus basse dans les joints de grain correspondent à la présence locale des unités de structures alternatives, alors que l'absorption ou l'émission des lacunes à un joint peuvent être regardées comme des transformations structurelles locales. Alors, quand la température monte et la concentration équilibre des lacunes monte, on augmente le désordre correspondant à la présence de diverses unités alternatives dans le joint. Ceci peut arriver ou graduellement, ou par une transformation du type ordre-désordre. On montre aussi que l'accumulation de lacunes ou d'impuretés dans le joint peut provoquer des transformations de glissement entre des structures de joint alternatives, conduisant en même temps à la migration des joints. Nous abordons ici l'importance de ces divers genres de transformations structurales pour les phénomènes qui se produisent dans les joints de grain. A number of non-equivalent structures have been found in the atomistic studies of periodic grain boundaries. These can transform into each other by removal or insertion of layers of atoms parallel to the boundary [19]. The ensuing multiplicity of structures is very extensive for long period, general, boundaries. We show here that low energy configurations of vacancies in grain boundaries correspond to the local presence of units of alternative structures so that absorption or emission of vacancies at a boundary can be regarded as local structural transformations. Hence, as the temperature rises and the equilibrium concentration of vacancies increases the disorder corresponding to occurrence of various alternative units in the boundary increases. This may happen either gradually or through an order-disorder transformation. It is further shown that accumulation of vacancies or impurities in the boundary may induce shear transformations between the alternative boundary structures which leads at the same time to the migration of the boundary. The significance of these different structural transformations for various grain boundary phenomena is discussed.
The methods available to study the correlation between properties and atomic structure of interfaces are critically assessed by considering the specific limitations. Studies of the behaviour of grain boundaries by means of the plate/sphere method are reported indicating that a few boundaries exhibit special properties which were observed for a particular interface independently of the property investigated (e.g. energy, corrosion, embrittlement). Most special boundaries were of high coincidence type. But not all high coincidence boundaries showed special properties. It seems the interatomic interaction which selects between special and "non-special" high coincidence boundaries. The properties of boundaries deviating from special misorientations were found to be controlled by the presence of (secondary) boundary dislocations. systematic studies of the properties of interphase boundaries between ionic crystals and noble metals showed that coincidence misorientations do not result in special behaviour which was observed only if close packed rows of atoms at the "surfaces" of the metal crystals "locked" into the "valleys" between close packed atomic rows at the "surfaces" of the ionic crystals ("lock-in" model of interphase boundaries). The basic idea of nanocrystalline materials is to generate a new type of solids by exploiting the highly distorted atomic structure existing in the core of grain (interphase) boundaries. This is achieved by reducing the crystal size of a polycrystalline material to a few nanometers (nanocrystalline material) so that the volume fraction occupied by interfaces (interfacial component) and crystals are comparable. The interfacial component of such a material consists of many (typically 1019 cm-3) boundaries. As the structures of these boundaries are all different, the interfacial component (i.e. the sum of all boundary structures) resembles a frozen gas. Experimental studies of nanocrystalline materials support this hypothesis and suggest that interfaces may be used as a structural component for generating a gas-like solid state structure.
Authors present calculations of surface tension (surface stress gamma ) for the body-centered cubic (b. c. c. ) metals V, Nb, Ta, Mo and W made using simple empirical N-body potentials obtained by M. W. Finnis and J. E. Sinclair. Results for several crystal faces are evaluated and compared with the corresponding surface energies sigma . Although there is a broad correlation between the calculated tensions and energies for different metals, and the two quantities are of a similar order of magnitude, they differ from one another by a factor of up to 2. Variant potentials for molybdenum were constructed to test the sensitivity of the results; negative surface tensions were found in some cases and shown to be a symptom of metastability of the b. c. c. structure.
The structure of symmetrical high-angle tilt boundaries with a 〈110〉 axis has been calculated using a potential for aluminium for 3 ⩽ ∑ ⩽ 19. In addition, since they have different symmetry, the (311) ∑ = 3. 1180° [1̄21] and (123̄) ∑ = 7. 38.21° [111] boundaries have been studied. The results were interpreted in terms of the formation of random close-packed groups and the symmetry properties of bicrystals.RésuméOn a calculé la structure de joints de flexion symétriques de forte désorientation et d'axe 〈 110〉 en utilisant un potentiel pour l'aluminium, pour 3⩽ ∑ ⩽19. On a également étudié les joints (311) ∑ = 3, 180 degrés/[1̄21] et (123̄) ∑ = 7, 38,21 degrés/ [111], dont la symétrie est différente. On interprète les résultats par la formation de groupes compácts aléatoires et par les propriétés de symétrie des bicristaux.ZusammenfassungDie Struktur einer symmetrischen Groβwinkelkorngrenze mit 〈 110〉 -Achse wird mit einem Potential für Aluminium für 3 ⩽ ∑ ⩽ 19 berechnet. Auβerdem wurden die Grenzen (311) ∑ = 3, 180°/[1̄21] und (123̄) ∑ = 7, 38,2°/ [111] studiert, da sie unterschiedliche Symétrie aufweisen. Die Ergebnisse werden anhand der Bildung statistisch dicht gepackter Gruppen und der Symmetrieeigenschaften von Bikristallen interpretiert.
The migration energies and atomic configurations for mono- and di-interstitials and mono- and divacancies in α iron have been calculated using a classical model. About 530 atoms surrounding the defect were treated as individual particles, each with three degrees of freedom, while the remainder of the crystal was treated as an elastic continuum with atoms imbedded in it. A two-body central force was devised which matched the elastic moduli, was sharply repulsive at close separation, and which went to zero midway between the second and third neighboring atoms. Configurations were found by choosing a starting configuration roughly approximating the situation under consideration and successively adjusting the value of each variable occurring in the energy equation so that the magnitude of the generalized force associated with it was zero until equilibrium was reached. The energy calculations include changes in bond energy in the discrete region, energy in the elastic field, and work done against cohesive forces, but neglect changes due to the redistribution of electrons. Calculated activation energies for motion of mono- and di-interstitials and mono- and di-vacancies were 0.33, 0.18, 0.68, and 0.66 eV, respectively, and binding energies of di-interstitials and di-vacancies were 1.08 and 0.20 eV, respectively. The stable interstitial was a "split" configuration in which two atoms were symmetrically split in a 〈110〉 direction about a vacant normal lattice site, and the stable di-interstitial consisted of two parallel split interstitials at nearest-neighbor lattice sites with their axes perpendicular to the line joining their centers. In the vacancy configuration an atom was missing from a normal lattice site, and the divacancy consisted of two vacancies at second-nearest-neighbor lattice sites.
The recently published semi-empirical potentials of Finnis and Sinclair for the metals V, Nb, Ta, Mo and W appear to give unphysical results for properties involving small interatomic separation. This is remedied by adding to the potentials cores fitted to electron gas calculations on dimers. The adjusted potentials are shown to predict a more realistic pressure-volume relationship. Interstitial formation energies are calculated for various configurations, using quenched molecular dynamics and static relaxation. Some preliminary results on interstitial migration are presented.
Recent calculations on coincidence twist grain boundaries in oxides with the rocksalt structure have shown that the interfaces are only weakly bound. This is in contrast with experiment in which twist grain boundaries have been observed in MgO. The static lattice relaxation calculation presented here predicts a restructured stable [001] twist boundary in NiO. The results are equally applicable to the other oxides with the same structure.
Structures and energies have been calculated for periodic, symmetrical left bracket 001 right bracket and left bracket 110 right bracket tilt boundaries in bcc metals using a variety of central-force potentials. The structures were found to be independent of the potential used and in general more than one relaxed structure was found for each type of grain boundary studied. The boundary structures are analyzed in terms of polyhedral groups, called compact polyhedra, with edge lengths which vary between the first- and second-nearest-neighbor separations. A smooth transition occurs between the low-angle and high-angle regimes. Relative to corresponding fcc boundaries, local relaxations were found to be at least as important as rigid translations and atoms were often found to occupy coincidence sites in minimum energy bcc boundaries.
A simple form of multi-ion interaction has been constructed for the purpose of atomistic simulation of transition metals. The model energy consists of a bonding term, which is the square-root of a site density ρi, summed over atoms i, and a repulsive pairwise term of the form The site density ρi is defined as sum over neighbouring sites j of a cohesive potential (R ij). Both V and are assumed to be short-ranged and are parameterized to fit the lattice constant, cohesive energy and elastic moduli of the seven body-centred-cubic (b.c.c.) transition metals. The result is a simple model which, unlike a pair-potential model, can account for experimental vacancy-formation energies and does not require an externally applied pressure to balance the “Cauchy pressure”.
The N-body potentials of Finnis and Sinclair (1984) are investigated to gauge their usefulness in computer simulation work. Results on monovacancies, di-vacancies and (100) surfaces are presented.
Relaxation volumes of point defects in cubic metals can be conveniently estimated by differentiating approximate formation energies with respect to the lattice constant. Lowest-order analytical expressions for monovacancies have been derived for equilibrium pair potentials and for the N-body potentials of Finnis and Sinclair (1984). These expressions may be evaluated with little effort and prove far superior to other lowest-order approximations to vacancy relaxation volumes.
A new, semiempirical model of metals and impurities (embedded atom method) makes possible a static treatment of the brittle fracture of a transition metal in the presence of hydrogen. Results indicate that hydrogen can reduce the fracture stress in nickel.
The structure of grain boundaries can be studied in atomic detail with the field ion microscope. A number of grain boundaries in tungsten and tungsten-rhenium alloys have been examined and the information from these studies has been collected together for presentation in this paper. The distribution of strain energy in the region of a low angle boundary leads to preferential field evaporation of material from this region. The field ion microscope has proved more useful for elucidating the atomic structure at high angle grain boundaries. The width of high angle boundaries is observed to be very narrow (approximately 2 atom diameters) and such a boundary shows regions of good and bad fit. These observations are correlated with a model for high angle boundaries, presented in this paper and developed from coincidence site lattice theory. For two grains related by particular misorientations about specific axes, high density coincidence site lattices exist and these special relationships are tabulated. It is also shown that deviations from these relationships can be accommodated by the introduction of a dislocation sub-boundary. The effect of the orientation of the boundary plane is also given and it is shown that good fit regions are developed where the boundary follows the most densely packed planes in the coincidence lattice. A stepped structure is generated where the boundary runs at an angle to these planes. Segregation effects at grain boundaries are illustrated and it is shown that the processes giving rise to an image from a finite concentration solid solution make it difficult to determine the atomic structure of a grain boundary in such a material.
The energies and motions of grain boundaries between two crystallites are investigated theoretically using the dislocation model of grain boundaries. Quantitative predictions made for simple boundaries for cases in which the plane of the boundary contains the axis of relative rotation of the grains appear to agree with available experimental data. The quantitative expression for energy per unit area for small angles is approximately [Ga / 4Ï(1-Ï)]θ[A-lnθ] where G is the rigidity modulus, a the lattice constant, Ï Poisson's ratio, θ the relative rotation and A approximately 0.23. Grain boundaries of the form considered may permit intercrystalline slip and may act as stress raisers for the generation of dislocations.
A review of geometric criteria for low interfacial energy which have been proposed in the literature is given. These include, (i) low reciprocal volume density of coincidence sites; (ii) high planar density of coincidence sites, [Gamma]; (iii) high [Gamma] at constant interplanar spacing, d; (iv) large d; and (v) high density of locked-in rows of atoms. These criteria are then tested against available experimental results which include measurements of: (a) interfacial energy; (b) rotations of crystallites on flat crystal substrates; (c) boundary faceting; (d) boundary dissociation; and (e) observations of grain boundary dislocations. No support for the general usefulness of criteria (i), (ii), (iv) and (v) is found. In all cases, significant numbers of results violating these criteria are found or else it is demonstrated that their range of validity is undefined, and, hence, their predictive power is highly limited. Criterion (iii) is found to apply for a limited number of cases involving metal/metal or ionic/ionic interfaces but fails for metal/ionic interfaces. Further testing and consideration of this criterion seems called for. It is concluded that no general and useful criterion for low energy can be enshrined in a simple geometric framework. Any understanding of the variations of interfacial energy must take account of the atomic structure and the details of the bonding at the interface.
A model potential-energy function comprising both two- and three-atom contributions is proposed to describe interactions in solid and liquid forms of Si. Implications of this potential are then explored by molecular-dynamics computer simulation, using 216 atoms with periodic boundary conditions. Starting with the diamond-structure crystal at low temperature, heating causes spontaneous nucleation and melting. The resulting liquid structurally resembles the real Si melt. By carrying out steepest-descent mappings of system configurations onto potential-energy minima, two main conclusions emerge: (1) a temperature-independent inherent structure underlies the liquid phase, just as for ``simple'' liquids with only pair interactions; (2) the Lindemann melting criterion for the crystal apparently can be supplemented by a freezing criterion for the liquid, where both involve critical values of appropriately defined mean displacements from potential minima.
Interfacial Structure, Properties and Design
- S R Phillpot
- D Wolf