This content has been downloaded from IOPscience. Please scroll down to see the full text.
Download details:
IP Address: 134.151.40.2
This content was downloaded on 16/01/2014 at 14:25
Please note that terms and conditions apply.
An embedded atom method potential of beryllium
View the table of contents for this issue, or go to the journal homepage for more 2013 Modelling Simul. Mater. Sci. Eng. 21 085001
(http://iopscience.iop.org/0965-0393/21/8/085001)
Home Search Collections Journals About Contact us My IOPscience
Modelling Simul. Mater. Sci. Eng.21(2013) 085001 (14pp) doi:10.1088/0965-0393/21/8/085001
An embedded atom method potential of beryllium
Anupriya Agrawal1,3, Rohan Mishra1,4, Logan Ward1,5, Katharine M Flores2and Wolfgang Windl1
1Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43201, USA
2Department of Mechanical Engineering and Materials Science, Washington University, St. Louis, MO 63021, USA
E-mail:windl.1@osu.edu
Received 13 February 2013, in final form 11 September 2013 Published 16 October 2013
Online atstacks.iop.org/MSMSE/21/085001 Abstract
We present an embedded atom method (EAM) potential for hexagonal beryllium, with a pair function in the form of a Morse potential and a Johnson embedding function with exponential electron density. The cohesive energy, elastic constants, lattice parameters and relaxed vacancy formation energy were used to fit the potential. The fitted-potential was validated by a comparison to first-principles and, wherever available, experimental results for the lattice energies of various crystal structures: vacancy cluster, interstitial formation and surface. Using a large cutoff distance of 5 Å, which includes interactions to approximately the eighth neighbor shell of beryllium, allows our potential to reproduce these quantities considerably better than previous EAM potentials.
The accuracy obtained by our potential is similar to or in some cases even better than available modified EAM potentials, while being computationally less intensive.
(Some figures may appear in colour only in the online journal)
1. Introduction
The exceptionally low density of beryllium, its moderate strength, high thermal stability and very high elastic modulus make it an appealing choice for many aerospace structural applications. Its low density and atomic mass also make beryllium highly transparent to x-rays, making the material ideal for radiation windows of x-ray tubes and other particle-physics equipment [1]. In non-elemental form, the addition of beryllium to copper significantly
3Present address: Department of Chemistry, Clemson University, Clemson, SC, 29634, USA.
4 Present address: Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA;
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA.
5 Present address: Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208–3108, USA.
increases the strength and toughness of the resulting alloy, enabling applications in various types of tools such as springs, pliers, hammers, chisels, etc [2]. Since the early 1990s, Johnson et alhave developed a series of beryllium-containing amorphous metallic alloys, which exhibit improved processability and glass-forming ability [3,4]. However, the toxic behavior and high cost of beryllium limit its widespread use. Better prediction and optimization of the properties of beryllium-based alloys through computational modeling are therefore necessary to identify applications where the benefits in improved performance outweigh the economic costs of the material.
To further the understanding of the properties of beryllium and its alloys for this wide variety of applications, experiments and simulations are equally important tools. Atomistic simulations using classical molecular dynamics with empirical potentials can be performed at very reasonable computational costs to better understand the behavior of beryllium in different environments. In order to do that, good-quality potentials are necessary that reproduce structure and atomic scale mechanisms in beryllium reasonably well. In addition, recent advances in alloy-potential generation, where high-quality multi-component potentials are produced on the basis of accurate elemental potentials, [5] make their generation even more desirable. However, a good potential, which could describe the properties of beryllium, is still unavailable. In this work, we report a embedded atom method (EAM) potential of beryllium, which is the best of all the available potentials of beryllium.
The era of interatomic potentials with predictive power for metals began with the advent of the EAM developed by Daw et al [6–8]. EAM potentials have been instrumental in developing an understanding of the deformation mechanisms of metals through atomistic simulations. EAM potentials have been used extensively in studying the elastic properties, vacancy formation energies, dislocation movement and fracture and fatigue mechanisms of metals. Later extensions include the modified embedded atom method (MEAM) [9], where angular forces are introduced at the expense of including three-body terms to allow for the simulation of directional bonding, important for non-metals (e.g. Si) and some metal alloys.
However, the addition of the three-body term makes the MEAM potential development more time consuming and their use computationally more expensive.
Prior work to develop interatomic potentials for beryllium has been sparse. Karimiet al reported a beryllium EAM potential which predicts the face centered cubic (fcc) structure to be more stable than the experimentally observed hexagonal closed packed (HCP) structure [10].
Igarashiet alreported Finnis–Sinclair [11] potentials for all HCP metals including beryllium, which predict elastic properties well, but produce almost three times higher interstitial formation energies than theab initiovalues [12]. Pasianotet alargued that the high interstitial formation energies are a result of neglecting the effect of the internal degrees of freedom.
They also contended that it was impossible to develop an EAM potential for HCP metals for which (a) (3C12−C11)/2 > C13−C44 and (b)C13−C44 >0 do not hold true. Therefore, they did not develop potentials for yttrium, cadmium, zinc, zirconium and beryllium [13].
Luoet al[14] developed Sutton-Chen type many-body potentials for beryllium in which they included zero-point energy in the potential optimization. This potential was able to predict the melting temperature of beryllium to be 1350 K from two-phase simulations. However, the validity of this potential for calculating phase stability, defect energies, etc were not reported.
Hcp beryllium has ac/a ratio of 1.568, which is 4% lower than the idealc/a value of 1.63; this suggests that bond directionality is present in beryllium. Due to the presence of a combination of the metallicity and directionality in beryllium, it is rather difficult to develop its EAM potential. Since EAM potentials are non-directional, a MEAM potential may, in principle, be more suitable for beryllium. Indeed, Baskes and Johnson [15] developed a MEAM potential for beryllium that included the effect of angular forces, which represented the lattice
2
parameters and the cohesive energy of beryllium in good agreement with experimental values.
However, this potential leads to a phase change at 100 K and therefore is unsuitable for finite temperature MD simulations [17]. Huet al developed a modified analytic EAM potential that had an additional energy term to express the difference between the actual total energy of a system of atoms and that calculated from the original EAM using a linear superposition of spherical atomic electron densities [16]. This potential predicted the formation energies of the various types of self-interstitials to be the same, which contradicts the first-principles results. Thompsonet al[17] developed yet another MEAM potential by taking fcc beryllium to be the reference state. Although this potential results in a better melting temperature (of 1390 K), it gives ac/aratio of 1.62, which is very close to the ideal value. Dremovet al[18]
also developed a MEAM potential with HCP beryllium as the reference state, which predicts ac/aratio of 1.60 [17]. Thus, directionality alone is not enough for a potential that describes beryllium and its properties physically correctly with adequate numerical accuracy, although the proposed MEAM potentials outperform the previous EAM potentials considerably.
Given the shortcomings of the previous EAM potentials for beryllium, what is still missing is a good-quality EAM potential that predicts its properties qualitatively correctly and is as quantitative as possible, which is particularly needed for use in alloy potentials. Such an elemental potential can then be incorporated into alloy potentials, such as the RAMPAGE framework by Wardet al[5] which generates multi-component potentials from high-quality elemental EAM potentials by fitting Finnis–Sinclair type cross potentials [11] toab initiodata for the binary intermetallics. In this paper, we report an EAM potential for beryllium, which we find to be considerably more successful in reproducing the experimental properties than the previously reported EAM potentials.
2. Theory of EAM potential
For an elemental metal, the total energy of a system consisting ofN atoms can be written as Etot =
N
i=1
Ei, (1)
whereEiis the energy of theith atom.
In EAM,Ei is defined as [6–8]
Ei = 1 2
i=j
φ (rij)+F (ρi), (2)
whererijis the interatomic distance between two atomsiandj.φis the pair-interaction energy, which is a function of the interatomic distance between atomsiandj. F is the embedding function, which depends on the electron densityρi, given by
ρi=
i=j
f (rij), (3)
whereρi is the sum of the electron densitiesf contributed by neighbouring atomsj at the position of atomi, that are functions of the interatomic distancesrij. The embedding function represents the energy associated with embedding atomiinto the host electron density. To make the calculations fast without compromising accuracy, all functions are typically truncated at a cutoff radius, which is generally taken as the third nearest neighbour distance or larger.
The pair functionφhere is modeled as a Morse potential given by
φ=De(e−2α(rij−re)−2e−α(rij−re)), (4)
whereDe is the cohesive energy, r is the interatomic distance, re is the nearest-neighbor equilibrium bond length andαis a measure of the curvature at the minimum of the potential curve. The Morse potential has been found in previous EAM work to give good accuracy [19], while the small number of three parameters facilitates fitting [20]. To avoid any discontinuity in the pair function and its derivatives atrcutoff, the ‘Voter taper’ is applied in the form [19]
φtapered=φ (r)−φ (rcutoff)+rcutoff m
1−
r
rcutoff
m
dφ dr
rcutoff
, (5)
wheremis a constant. This taper function has an effect over the full range of the potential to ensure the continuity of the pair function and electron density function.
We chose the electron density functionf to be given by the commonly used exponential function [21]
f =Ae−B(rij−re), (6)
whereAandBare constants. To truncate the interaction range, we also apply a taper function equivalent to equation (5) to the electron density.
For the embedding function, we chose the Johnson function [22], which is given by
F =F0[1−ln(x)]x−F1y. (7)
In the above equation,F0andF1are constants,x =(ρi/ρo)β andy =(ρi/ρo)γ. ρo,β and γ are fitting constants. Because of its form and five fitting parameters, equation (7) allows considerable flexibility for fitting an adequate beryllium potential. It is important to note that Aandρonever appear individually but only as anA/ρoratio. Thus, their individual value is of no consequence. In this paper, we have therefore used a reduced electron density function, which isf/A, and the reduced electron densityρ/A.
2.1. Fitting procedure
We used the general utility lattice program (GULP) [23,24] for the potential fitting. The quality of the fit is monitored by a weighted sum of square residues,
G=
N
i=1
wi[fi(calc)−fi(obs)]2, (8)
whereiis the number the observables,wiis the weight associated with observableiandfi(calc) andfi(obs)are the calculated andab initio/experimental values of the observable, respectively.
To minimizeG, GULP uses a Newton–Raphson minimization approach for fitting the potential.
The best approach we found was to start fitting with only a few parameters and then increase their number gradually in subsequent restarts. For fitting, we used two structural configurations.
The first structure is a 256-atom simulation box with a HCP structure and a box size of 9.08 Å×15.68 Å×14.24 Å. The elastic constants matrix, cohesive energy, bulk modulus and shear modulus of this structure were used at the start of the fitting. The internal degrees of freedom, which allow the relaxation of internal coordinates, are included in GULP for the calculation of elastic constants. The second structure used in fitting is the same cell, but with 255 atoms and one vacancy. The relaxed vacancy formation energy fromab initio calculations, as discussed in the following section, was used as the fitting observable for this structure. The lattice parameters were kept constant at the start of the fitting and relaxed later.
All the observables and their corresponding weights used in fitting are given in table1.
4
Table 1.Parameters, their values and their weights (w) used in fitting.Ecis the cohesive energy, Cij are the elements of the stiffness tensorK,Gare the bulk and shear moduli andEf is the monovacancy formation energy.
Value EAMa EAMb MEAMc MEAMd MEAMh MEAMi
used in Error Error Error Error Error Error
Parameter fitting w Value (%) Value (%) Value (%) Value (%) Value (%) Value (%)
Ec(eV) 3.32e 5 3.34 0.7 3.70 11 3.43 3 3.32 0 — — 3.43 3
C11(GPa) 294f 4 291 −1 89 70 362 23 — — 259 −12 331 13
C33(GPa) 357f 4 357 0 257 28 193 −46 — — 329 −8 309 −13
C12(GPa) 27f 4 53 96 −59 −321 88 228 — — 77 185 −11 −141
C13(GPa) 14f 4 10 −28 −37 −361 −22 −260 — — 9 −36 18 29
C44(GPa) 162f 4 124 −24 107 −34 156 −4 — — 65 −60 19 −74
C66(GPa) 133f 4 119 −11 74 −44 137 3 — — 91 −32 171 29
K(GPa) 117f 4 121 3 116 −1 112 −5 — — 115 −2 113 −3
G(GPa) 150f 4 129 −14 153 2 137 −9 — — — — — —
Ef(1 V) (eV) 0.85g 4 1.26 48 1.13 33 1.23 45 1.12 32 — — — —
aCurrent work.
bReference [10].
cReference [15].
dReference [16].
eReference [31].
fReference [25].
gReference [32].
hReference [17].
iReference [18].
2.2. Ab initio calculations
We calculated energies of various (hypothetical) crystal structures of beryllium, as well as formation energies of di-vacancies, a number of self-interstitials and surface energies usingab initiocalculations for validation purposes. We used the Vienna Ab-initio Simulation Package (VASP) [26,27] to perform these calculations using projector augmented wave potentials [28]
within the generalized gradient approximation [29]. A plane-wave cutoff energy of 375 eV was used along with Monkhorst–Pack [30]k-point meshes to sample the Brillouin zone.
To calculate the cohesive energy of the HCP structure of beryllium, we took an orthogonalized [1 1 0]×[1 1 0]¯ ×[0 0 1] unit cell with dimensions 2.26 Å×3.92 Å×3.56 Å.
k-point meshes were selected such that the mesh divisionNifor each lattice vectoraiis given byNi =(40 Å)/ai. We also calculated the lattice parameters and cohesive energies of fcc, body centered cubic (bcc), simple cubic (SC) and diamond cubic (DC) structures of beryllium.
The calculated energies and lattice parameters along with experimental values [31] of the HCP structure are shown in table2. The experimental cohesive energy of beryllium was used for fitting to maintain the compatibility with potentials developed for other elements, where generally experimental values are used. This will allow the use of this beryllium potential as part of the alloy potentials.
We have also calculated formation energies of mono- and divacancy configurations (in- and out-of plane) (table3). These values match well with previously reportedab initiovalues [32].
For the case of self-interstitials, we studied the following eight different configurations in a cell size of 256 atoms: crowdion, tetrahedral, octahedral, basal crowdion, basal tetrahedral, trigonal, split interstitial0 0 0 1and basal split interstitial1 12 0¯ (as shown in figure 1).
We found that octahedral, basal crowdion, basal tetrahedral and basal split interstitials are unstable and relax to a trigonal interstitial, which we find to be the most stable interstitial overall. Although the trigonal interstitial has been identified previously to have the lowest
Table 2.Lattice constants and energies of different crystal structures of beryllium.
Ab initio
Structure Parameter Calc.a Expt.b EAMa EAMc MEAMd MEAMe MEAMf MEAMg
HCPEHCP a(Å) 2.27 2.286 2.363 2.290 2.286 2.286 — 2.243
c/a 1.568 1.568 1.568 — 1.568 1.568 1.620 1.600
eV/atom 3.727 3.320 3.226 3.700 3.430 3.320 — 3.430
FCCEFCC–EHCP a (Å) 3.16 — 3.198 — — — —
eV/atom 0.08 — 0.016 — 0.057 0.006 — —
BCCEBCC–EHCP a (Å) 2.50 — 2.546 — 2.568 — —
eV/atom 0.10 — 0.045 — 0.074 0.031 — —
Simple cubicESC–EHCP a(Å) 2.18 — 2.224 — 2.20 — — —
eV/atom 0.99 — 0.421 — 0.32 — — —
Diamond cubicEDC–EHCP a(Å) 4.90 — 5.038 — 5.03 — — —
eV/atom 1.59 — 0.776 — 1.68 — — —
aCurrent work.
bReference [31].
cReference [10].
dReference [15].
eReference [16].
fReference [17].
gReference [18].
formation energy, earlier work has not identified the instability of octahedral and basal split interstitials [32]. Also, while our interstitial formation energies in general agree reasonably well with the previous work [32], we find for the tetrahedral interstitial, a formation energy which is nearly 1 eV lower. In addition, we also calculated the surface energy of the (0 0 0 1) surface. All resulting energies are summarized in table3.
3. Validation and discussion
Several fits were attempted with cutoff radii ranging from 4.0 to 5.0 Å. The fitting quality was observed to improve on increasing the cutoff radius and consequently by including more neighboring atoms. The best fit was obtained for a cutoff radius of 5.0 Å, which includes the eighth nearest-neighbor shell for equilibrium lattice constants, while larger cutoff radii were not attempted to maintain computational efficiency. This observation was also made by Igarashiet alfor other HCP metals, where they chose cutoff radii between the sixth and eight nearest neighbor distances [12]. The three constituent functions used in the EAM potential, i.e. the pair function, the electron density function and the embedding function, are plotted in figure2. The Morse potential describing the pair interactions (equation (4)) has its minimum at the nearest neighbor distance of 2.29 Å (figure2(a)). As shown in figure2(a), the taper function, which is applied to full range of the pair potential, (equation (5)) significantly alters the pair function. This means that the untapered Morse function cannot be used by itself as a pair function for this particular potential. The fitted electron density as a function of the radial distance is shown in figure2(b) and the embedding function is shown in figure2(c).
Table4lists all fitting parameters and their values. Our embedding function has a minimum at a (reduced) density of∼7.7. However, at the reduced equilibrium electron density (ρe∼12.5), the embedding function is increasing while its slope is decreasing. This makesF(the second derivative of the embedding functionF with respect toρatρe)6negative and has an important consequence on stabilizing the beryllium HCP structure, as discussed later.
6 F= ddρ2F2
ρe
. 6
Table 3.Formation energies (in eV) of relaxed defects (both ion positions and lattice vectors) in the HCP structure. Unstable defects are indicated by the defect into which they decompose, where that information is available.
Ab initiob EAMb
256 256 16384
Defect configuration Ab initioa atoms atoms atoms EAMc EAMd MEAMe MEAMf
Monovacancy 0.85 0.81 1.28 — 1.13 1.10 1.23 1.12
Divacancy aa 1.96 1.84 2.33 — — — 2.29 2.09
Divacancy ca 2.07 1.95 2.45 — — — 2.25 2.11
Crowdion (c) 4.39 4.35 4.76 4.76 — 10.50 — bs
Tetrahedral (t) 5.22 4.31 4.69 4.69 — 12.29 — bs
Octahedral (o) 5.24 tr 4.78 4.78 — 10.61 — bs
Trigonal (tr) 4.20 4.09 4.99 4.99 — 11.87 — bs
Basal tetrahedral (bt) Unstable tr 5.15 5.15 — 11.55 — bs
Basal crowdion (bc) Unstable tr bs bs — 10.61 — bs
Split [0 0 0 1] (s) 5.29 5.20 5.31 5.31 — 12.18 — bs
Basal split [1 1 2 0] (bs) 4.30 tr 4.69 4.69 — — — 2.96
Surface Energy (0 0 0 1) (J m−2) — 1.72 2.16 — — — 1.65 1.27
aReference [32].
bCurrent work.
cReference [10].
dReference [12].
eReference [15].
fReference [16].
Figure 1. Different interstitials studied in the HCP structure of beryllium. (a) Split in [0 0 0 1]
direction, (b) basal split in [1 1 2 0] direction, (c) crowdion, (d) tetrahedral, (e) octahedral, (f) trigonal, (g) basal tetrahedral and (h) basal crowdion configurations [32].
For validation of our potential, we calculated the physical and mechanical properties of the HCP structure, the stability of other crystal structures relative to the HCP structure, as well as vacancy and interstitial formation energies and the basal surface energy using two common molecular dynamics codes, LAMMPS [33] and GULP [23].
One possible drawback of our potential could be the increase in the computational execution time because of the high cutoff radius. In order to examine its computational
Figure 2. The (a) pair potential, (b) reduced electron density and (c) embedding functions used for our EAM potential. The red (solid) line in (a) shows the bare pair function, the blue (dashed) line shows the tapered pair function with the Voter taper.
Table 4.EAM potential parameters and values obtained after fitting.
Function Parameters Values
Pair potential function De(eV) 0.412 46 α(Å−1) 0.363 24
re(Å) 2.290 00
Electron density function A/ρo(unitless) 1.597 00 B(Å−1) 0.497 13 Embedding function Fo(eV) 2.039 30 F1(eV) −12.6178 β(unitless) 0.187 52 γ(unitless) −2.288 27 Voter function m(unitless) 10.0000
Cutoff radius rcutoff(Å) 5.0000
efficiency, we compared the execution time for a molecular dynamics simulation of the beryllium HCP structure between our EAM potential and the MEAM potential by Baskes et al[15] as implemented in GULP [23]. Following the benchmarking recipe proposed on the LAMMPS [34] website, we find for our potential, which has on average 164 neighbors/atom, an average execution time of 4.01×10−6CPU seconds per timestep per atom. For the MEAM potential with on average 70 neighbors/atom, execution takes 1.83×10−5CPU seconds per timestep per atom, which is∼4 times higher than for our potential. Thus our potential still maintains the edge in computational efficiency that is typically expected from EAM potentials.
For the crystal structures listed in table2, lattice parameters and cohesive energies were calculated using the developed potential. A comparison of these values with theab initio data and MEAM is shown in table2. The DC structure was found to be least stable whereas the HCP structure was found to have the lowest energy, which is in agreement with theab initiocalculations (table2). The cohesive energyEcHCP, calculated fromab initiowith a value of 3.73 eV, is larger than the experimental value (3.32 eV) by 0.41 eV. This overestimation is typical for density functional theory (DFT) calculations [35]. For the HCP structure, the lattice parametersa calculated with our new potential is 2.36 Å, which agrees within 3% with the experimental value of 2.29 Å. However, thec/aratio that we obtained (1.568) is equal to the experimental value of 1.568 (table2). Our EAM potential accurately predicts the sequence of the energies of the different phases, with values comparable to the MEAM potential by Baskes and Johnson [15]. Both potentials not only reproduce the correct order of energies for
8
Figure 3. Energy variation with the change in volume of HCP (triangle), fcc (square) and bcc (circle) structures using our EAM potential (open symbols) and DFT calculations (filled symbols).
the different structures, but have similar average absolute errors of 0.3 eV (EAM) vs. 0.2 eV (MEAM), with the largest absolute error for EAM being 0.68 eV for the case of DC and 0.64 eV for MEAM for SC (table2).
The stability of the HCP structure relative to other structures is also shown in figure3 where the energies of fcc, bcc and HCP structures were calculated as a function of the volume per atom using our EAM potential and DFT calculations. Hcp is the most stable structure with both our EAM potential and DFT predictions. With expanding volume, HCP remains the most stable structure, while under compression, HCP is stable for volumes larger than 90%
of the equilibrium volume, while for smaller volumes, fcc is found to be more stable, which contradicts DFT results and experimental findings [36], thus setting a limit to the useful range of our potential. This limitation of EAM potentials for HCP metals under extreme compression has been noted before by Igarashiet al[12], but still allows its use for the vast majority of relevant stress conditions.
The elastic constants for the HCP structure of beryllium are tabulated in table1. The agreement between EAM results and experiments is in general very good, especially in comparison to the previous EAM potential from [10] (comparison values in brackets), with an average absolute deviation of 12 GPa (60 GPa). ExcludingC12andC13, which have small absolute values of 27 and 14.0 GPa, and for which small absolute deviations of 16 (32) and 17 (23) GPa respectively result in large relative errors, our potential has a small relative error of 6.7% in comparison to the 27% from [10].
Previously, Pasianotet al [13], in their attempt at developing EAM potentials for HCP metals, proposed that the following two conditions needed to be satisfied to allow fitting of an HCP potential within the EAM framework: (1)C13−C44>0 and (2)C12−C66 > C13−C44. They argued that the sign of C13 −C44 depended on F, which had to be greater than zero for the lattice to be stable. However, looking at the experimental values of the elastic constants, the first condition does not hold true for beryllium, while the second condition is not satisfied for the case of zirconium. In apparent agreement with that, they were not successful at developing EAM potentials within the same framework for both beryllium and zirconium [13]. However, in 1995, Goldstein and Jonsson [37] were able to successfully develop a good-quality zirconium EAM potential by including the internal degrees of freedom in the fitting of the elastic constants, which had been ignored by Pasianot and Savino [13].
Similarly, for the case of beryllium, our chosen embedding function allows us to develop a highly functional EAM potential while not satisfying the first condition stated (C13−C44>0).
Igarashiet alhave also shown that the sign of (C12−C66)depends on the sign ofF[12]. As discussed earlier, ourFis less than zero, which thus makes both (C12−C66)and (C13−C44) less than zero. Any adverse effect on the lattice stability is not noted due to the negative
Table 5.Calculated and experimental values of three shear modulus and anisotropy factor ().
Experiment EAMa EAMb MEAMc MEAMd MEAMe
Parameter [25] Value Error (%) Value Error (%) Value Error (%) Value Error (%) Value Error (%)
Ca(GPa) 133 119 −11 74 −44 137 3 91 −32 171 29
Cb(GPa) 162 124 −23 107 −34 156 −4 65 −60 19 −88
Cc(GPa) 163 169 4 115 −29 154 −6 159 −2 144 −12
(GPa) −17 −8 54 −63 −280 93 658 −1 96 −2 86
aCurrent work.
bReference [10].
cReference [15].
dReference [17].
eReference [18].
value ofFsince the requirements for mechanical stability in HCP lattices are fulfilled by our potential, which are [C44 = 124 GPa]> 0, [C11 = 291 GPa]>[C12 = 53 GPa] and [C33(C11+C12)=122 808 GPa2]>[2C213=200 GPa2].
On comparing the elastic constants calculated from our potential with the experimental values, we find that our potential predictsC12 (53 GPa) to be twice the experimental value (27 GPa), whileC13andC44 are∼25% off. It would only be possible to get perfect elastic constants if we allowed bcc to be the most stable phase, which was also found by Goldstein and Jonsson for the case of Zr [37]. Despite the overestimation, the elastic constants calculated by our potential are considerably better in comparison to the previously reported values from the EAM potential by Karimiet al[10], the MEAM potential by Baskes and Johnson [15] and the MEAM potential by Dremovet al[18]. Elastic constants predicted by the Thompsonet al MEAM potential [17] are better than those from other potentials. The values of the elastic constants were not reported for the EAM potentials by Igarashiet al[12], Luoet al[14] and for the MEAM potential by Huet al[16].
We have also compared the shear elastic moduliCa,Cb,Cc and anisotropy factor, which give a better idea about the mechanical stability of a crystal [15] for different EAM and MEAM potentials (table5). The relation between the elastic moduliCa,Cb,Ccandis [15]
Ca =C66 Cb =C44
Cc= 16(C11+ 2C33+C12−4C13) = 13(C11−C33+C12−C13).
(9)
We find that our potential again performs better than the previous EAM potential with smaller errors as compared to the experimental values.
The formation energy of vacancies and vacancy clusters can be calculated as Ef(Vi)=E(Vi,BeN−i)−
N−i N
E(BeN), (10)
whereEf(Vi)is the formation energy of a monovacancy and divacancy for i = 1 and 2, respectively, E(Vi,BeN−i)is the energy of a cell withN atomic sites andivacancies and E(BeN)is the energy of a perfect beryllium cell withNatoms. Energies for different vacancy configurations were calculated using our EAM potential and are compared to the values from first-principles and previous potentials in table 3. Although the vacancy formation energy with our EAM potential is higher than the first-principles result by 50%, the disagreement is close to the range of results from the previous EAM [10,12] and MEAM potentials [14,15], which are respectively 33%, 29%, 45% and 32% higher than the first principles value. Our potential also shows the right trend when considering divacancy clusters, which predicts the
10
Figure 4.Different interstitial structures, after being relaxed using our EAM potential, in the HCP structure of beryllium with (a) split in [0 0 0 1] direction, (b) basal split in [1 1 2 0] direction, (c) crowdion, (d) tetrahedral, (e) octahedral, (f) trigonal, (g) basal tetrahedral and (h) basal crowdion configurations.
in-plane divacancy to have a lower energy than the out-of-plane divacancy in agreement with the first-principles results. Whereas the MEAM potential by Huet al[15] results in the same trend, the MEAM potential by Baskes and Johnson [15] predicts the out-of-plane divacancy to have a lower formation energy than the in-plane divacancy.
Different types of interstitials reported in the literature [32], as shown in figure1, were created in the beryllium HCP structure of 256 atoms. Their interstitial formation energy was calculated as
Ef(I )=E(I,BeN+1)− N+ 1
N
E(BeN), (11)
whereEf(I )is the interstitial formation energy andE(I,BeN+1)is the energy of the relaxed system containingNatoms at perfect crystal lattice sites and one self-interstitial atom. Energies of these configurations were calculated using density-functional theory and our EAM potential and are shown together with values from previous EAM/MEAM potentials in table3. Also, the structure of the different interstitials relaxed with our EAM potential is shown in figure4.
The relaxed interstitial formation energies calculated from our potential are in general in good agreement with our calculatedab initiovalues, but with some variations in the order of the different configurations. In particular, while the first-principles calculations identify the trigonal interstitial as the ground state, tetrahedral and basal split interstitial have the identical lowest energy with our EAM potential, whereas the trigonal interstitial is found to be 0.3 eV higher in energy. Comparing the initial and relaxed interstitials structures (figures1(h) and4(h)), it is evident that our potential predicts the basal crowdion interstitial to be unstable, as it was within DFT, and relaxes to a basal split configuration. Overall, the EAM values are higher than the first-principles values by, on average, 14%. In contrast, the EAM potential developed by Igarashiet al [12] predicts the crowdion interstitial to be the most stable and calculates values that are throughout more than twice as large as the first-principles results with an average error of 165%, making it an unfavorable choice for simulating defects. The MEAM potential by Huet al[15] reported an interstitial formation energy 1/3 smaller than the corresponding first-principles value and predicts just one stable interstitial configuration, which is basal split, while all other configurations relax to it. Summarizing these comparisons,
Figure 5. Volume versus temperature curves for HCP beryllium. The experimental data is from [38]. The difference in volume is discussed in the text.
the potential developed in this work seems to be advantageous over those previous potentials especially with respect to interstitial formation energies and thus should be better suited for studying radiation damage or complex defect energies. In order to check the convergence of the interstitial formation energy with cell size, we have repeated the calculations with a much larger 16384 atoms cell and find the formation energies to be similar to a 256 atoms cell, as shown in table3.
Finally, we calculated the Be (0 0 0 1) relaxed surface energy using DFT as well as our EAM potential. The surface energies obtained are 1.72 J m−2(ab initio) and 2.16 J m−2(EAM). The energy obtained by our potential is 25% larger. This compares well with the MEAM potentials, which underestimate the first-principles value by 4% [14] and 26% [15], respectively.
To test the temperature dependent properties of beryllium, we studied the expansion of beryllium within the temperature range 300–1100 K. These simulations were carried out in anNPT ensemble, where the temperature was increased in steps of 20 K within 0.1 ps and equilibrated at each temperature for 3 ps. We observe that the thermal expansion of beryllium is not linear below the melting temperature. As a result, the thermal expansion coefficient is not constant in this temperature range, as shown in figure5. Experiments carried out by Gordon et al[38] confirmed that the thermal expansion of beryllium is not linear below the melting temperature. More recently, Lazickiet al[36] also emphasized this behavior of beryllium. The volume difference between the simulated and experimental curves is due to our higher values ofa(2.36 Å)andc(3.70 Å)than their experimental values of 2.29 Å and 3.58 Å, respectively.
Also, we predict the volume coefficient of thermal expansion higher than experimentally found, from 47% at 400 K to 75% at 1100 K in the range displayed in figure5.
We used the two-phase simulation within theNPTensemble suggested by Belonoshko and Dubrovinksy [39] to calculate the melting temperature of beryllium. Starting from a system with 160 000 atoms, wherein one-half of the atoms are in the liquid phase and the other-half are in the solid phase (figure6(a)), we equilibrate it at zero pressure and different temperatures, allowing the system to go to a single phase. While at 1200 K, everything becomes crystalline (figure6(b)), at 1300 K, the system goes to the amorphous phase (figure6(c)). Thus, the estimated melting temperature of beryllium using our potential is between 1200–1300 K, which is slightly below the experimental value of 1550 K [25]. Dremovet al[17] were not successful in using the two-phase method to calculate the melting temperature, however, from the single-phase method, they found the melting temperature at 0 GPa pressure to be 1550 K, which is in excellent agreement with experiments. It is important to note that the single-phase
12
Figure 6.Snapshots of a system of 160 000 atoms in (a) the two-phase configuration used as the initial state to determine the melting temperature of Be, (b) crystalline phase formed after 0.5 ps of simulations at 1200 K and (c) amorphous phase, formed after 0.5 ps of simulation time at 1300 K.
MD method is known to predict a higher melting temperature, as reported by Luoet al[14].
Thompsonet al[17] predicted the melting temperature of beryllium to be 1390 K using the two-phase method.
4. Conclusions
We have presented an empirical potential of beryllium in a simple EAM format, which is easy to implement and appropriate for developing cross-potentials for studying multi-component systems. The potential developed compares quite favorably with previous EAM and MEAM potentials. Our potential predicts the lattice stability as well as the relative stabilities of various lattices correctly. When including internal degrees of freedom, the elastic constants calculated with our potential are positive and follow one of the two essential criterions laid out by Pasianot and Savino [13],C12−C66 > C13−C44. Additionally, our potential correctly predicts that C12−C66<0 andC13−C44 <0, which is due to the curvature of the embedding function,F, being negative, as shown previously [12,13]. The lattice constants and cohesive energy for the HCP structure are predicted within 1% of the experimental values. We also calculated monova- cancy and divacancy formation energies for which we find values that are 50% and 25% higher than theab initiovalues, respectively, but are comparable to the results from previous EAM and MEAM potentials. Finally, we have calculated self-interstitial formation energies for various configurations and have obtained energies within∼14% of the first-principles results, which is a considerably better agreement than found for previously published values from EAM or MEAM potentials. Due to the favorable prediction of defect formation energies, our EAM potential should be highly appropriate for simulating radiation damage and related problems. High strain simulations can also be carried out using our potential since the HCP phase remains stable at high strains. The HCP phase also remains stable when carrying out molecular dynamics simula- tions at high temperatures with our potential, which was a major disadvantage of the MEAM po- tential in [15]. The non-linear thermal expansion behaviour of beryllium observed experimen- tally is reproduced well with the developed potential. Our potential also gives a melting tem- perature between 1200 and 1300 K, as compared to the experimental value of 1550 K. As a final note, the potential generated is available for download on the internet in LAMMPS format [40].
Acknowledgments
This work was supported by the Air Force Office of Scientific Research under grant numbers FA9550-09-1-0251 and FA9550-12-1-0059. RM and WW acknowledge partial support from the Center for Emergent Materials at the Ohio State University, an NSF MRSEC under grant number DMR-0820414. WW and KMF acknowledge partial support from the Defense Threat Reduction Agency under project number HDTRA1-11–0047. LW was supported by the Department of Defense (DoD) through the National Defense Science and Engineering Graduate Fellowship (NDSEG) Program. All of the calculations were performed at the Ohio Supercomputing Center (Grant No PAS0072).
References
[1] Veness R, Ramos D, Lepeule P, Rossi A, Schneider G and Blanchard S 2009Particle Accelerator Conf.
(Vancouver, May 2009)M06RFP010 [2] www.globalsecurity.org/wmd/intro/beryllium.htm [3] Peker A and Johnson W L 1993Appl. Phys. Lett.632342
[4] Hofmann D H, Suh J-Y, Wiest A, Guan G, Lind M-L, Demetriou M D and Johnson W L 2008Nature4511085 [5] Ward L, Agrawal A, Flores K M and Windl W 2009 arXiv:1209.0619[cond-mat.mtrl-sci]
[6] Daw M S and Baskes M I 1984Phys. Rev.B296443
[7] Foiles S M, Baskes M I and Daw M S 1986Phys. Rev.B337983 [8] Daw M S, Foiles S M and Baskes M I 1993Mater. Sci. Rep.9251 [9] Baskes M I 1987Phys. Rev. Lett.592666
[10] Karimi M, Yang Z, Tibbits P, Ila D, Dalins I and Vidali G, 1990MRS Proc.19383 [11] Finnis M W and Sinclair J E 1984Phil. Mag.A5045
[12] Igarashi M, Khantha M and Vitek V 1991Phil. Mag.B63603 [13] Pasianot R and Savino E J 1992Phys. Rev.B4512704
[14] Luo S-N, Ahrens T J, Cagin T, Strachan A, Goddard W A and Swift D C 2003Phys. Rev.B68134206 [15] Baskes M I and Johnson R A 1994Modelling Simul. Mater. Sci. Eng.2147
[16] Hu W, Zhang B, Huang B, Gao F and Bacon D J 2001J. Phys.: Condens. Matter131193 [17] Thompson A P, Lane J M D, Desjarlais M P and Baskes M I 2009AIP Conf. Proc.1195833
[18] Dremov V V, Karavaev A V, Sapozhnikov F A, Vorobyova M A and Soulard L 2009AIP Conf. Proc.1195837 [19] Voter A F 1993Los Alamos Unclassified Technical Report#LA-UR 93–3901
[20] Abraham R J and Stolevik R 1978Chem. Phys. Lett.58622 [21] Johnson R A 1988Phys. Rev.B373924
[22] Johnson R A 1989Phys. Rev.B3912554 [23] Gale J D 1997JCS Faraday Trans.93629 [24] Gale J D and Rohl A L 2003Mol. Simul.29291
[25] Migliori A, Ledbetter H, Thoma D J and Darling T W 2004J. Appl. Phys.952436 [26] Kresse G and Hafner J 1993Phys. Rev.B47558
[27] Kresse G and Hafner J 1994Phys. Rev.B4914251 [28] Blochl P E 1994Phys. Rev.B5017953
[29] Wang Y and Perdew J P 1991Phys. Rev.B4413298 [30] Monkhorst H J and Pack J D 1976Phys. Rev.B135188 [31] Chou M Y, Lam P K and Cohen M L 1983Phys. Rev.B284179
[32] Ganchenkova M G, Vladmirov P V and Borodin V A 2009J. Nucl. Mater.38679 [33] Plimpton S 1995J. Comput. Phys.1171–19
[34] http://lammps.sandia.gov/bench.html
[35] Sholl D S and Steckel J A 2009Density Functional Theory: A Practical Introduction(Hoboken, NJ: Wiley) p 222
[36] Lazicki A, Dewaele A, Loubeyre P and Mezouar M 2012Phys. Rev.B86174118 [37] Goldstein A S and Jonsson H 1995Phil. Mag.B711041
[38] Gordon P 1949J. Appl. Phys.20908
[39] Belonoshko A B and Dubrovinksy L S 1996Am. Mineral.81303 [40] http://atomistics.osu.edu/
14