ATOMIC STRUCTURE OF THE S5 �310�=�001� SYMMETRIC

TILT GRAIN BOUNDARY IN MOLYBDENUM

G. H. CAMPBELL1{, J. BELAK2 and J. A. MORIARTY2

1Chemistry and Materials Science Directorate, University of California, Lawrence Livermore NationalLaboratory, Livermore, CA 94550, U.S.A. and 2Physics and Space Technology Directorate, University

of California, Lawrence Livermore National Laboratory, Livermore, CA 94550, U.S.A.

AbstractÐAtomistic simulations o�er an important route towards understanding and modeling materialsbehavior. Incorporating the essential physics into the models of interatomic interactions is increasingly dif-®cult as materials with more complex electronic structures than f.c.c. transition metals are addressed. Forb.c.c. metals, interatomic potentials have been developed that incorporate angularly dependent interactionsto accommodate the physics of partially ®lled d-bands. A good test of these new models is to predict thestructure of crystal defects and compare them with experimentally observed defect structures. To that end,the S5 �310�=�001� symmetric tilt grain boundary in Mo has been fabricated and characterized by HREM.The experimentally observed structure is found to agree with predictions based on atomistic simulationsusing angular-force interatomic potentials developed from model generalized pseudopotential theory(MGPT), but disagrees with predictions based on radial-force potentials, such as those obtained from theFinnis±Sinclair method or the embedded atom method (EAM). # 1999 Acta Metallurgica Inc. Publishedby Elsevier Science Ltd. All rights reserved.

Re sumeÂÐLes re sultats de simulations aÁ l'e chelle atomique permettent de mieux comprendre et de mode liserle comportement des mate riaux. Les justi®cations physiques sur lesquelles reposent les mode les d'interac-tions atomiques deviennent de plus en plus discutables lorsque des mate riaux posse dant une structure elec-tronique plus complexe que celle des me taux de transition cubiques aÁ faces centre es sont conside re s. Dansle cas des me taux cubiques centre s, des potentiels interatomiques avec de pendance angulaire ont e te devel-ope s de manieÁ re aÁ tenir compte de la physique des bandes d partiellement occupe es. Un des tests pertinantspour cette nouvelle mode lisation est de pre dire la structure de de fauts cristallins et comparer les re sultats aÁceux obtenus expe rimentalement. Dans ce but, le joint de grain aÁ inclinaison syme trique S5 �310�=�001� ae te produit expe rimentalement dans le cas de Mo et characte rise par microscopie e lectronique aÁ haute re s-olution. La structure observe e experimentalement est en accord avec les pre dictions de simulations qui utili-sent les potentiels interatomiques avec de pendance angulaire de rive s de la the orie des pseudopotentielsge ne ralise s, mais en de saccord avec les pre dictions obtenues aÁ partir de potentiels de paires, tels que ceuxde rive s de la me thode de Finnis±Sinclair ou de la me thode de l'atome entoure . # 1999 Acta MetallurgicaInc. Published by Elsevier Science Ltd. All rights reserved.

ZusammenfassungÐAtomistische Simulationen stellen ein wichtiges Mittel zur VerfuÈ gung, um das Verhal-ten von Materialien zu verstehen und zu modellieren. Das Einbauen der erforderlichen Physik in ModellefuÈ r zwischenatomare Wechselwirkungen wird progressiv schwieriger, wenn man Materialien mit elektro-nischen Strukturen behandelt, die komplizierter sind als kubisch ¯aÈ chenzentrierte UÈ bergangsmetalle. Umdie Physik der teilweise gefuÈ llten d-Niveaus zu beruÈ cksichtigen, wurden fuÈ r kubisch raumzentrierte MetallewinkelabhaÈ ngige zwischenatomare Potentiale entwickelt. Ein guter Test dieser neuen Modelle ist, die Struk-turen von Kristalldefekten vorauszusagen und mit experimentell beobachteten Strukturen zu vergleichen.Zu diesem Zweck wurden symmetrische S5 �310�=�001� Kippkorngrenzen in MolybdaÈ n hergestellt und ineinem Elektronenmikroskop mit hoher Au¯oÈ sung (HREM) charakterisiert. Die experimentell beobachteteStruktur stimmt mit der berechneten atomistischen Simulation uÈ berein, die winkelabhaÈ ngige zwischenato-mare Potentiale verwendet, welche vom Modell der Generalisierten Pseudopotential Theorie hergeleitetsind. Sie stimmt jedoch nicht uÈ berein mit Berechnungen, die Potentiale verwenden, welche lediglichvom zwischenatomaren Abstand abhaÈ ngen, wie beispielsweise solche, die von der Finnis±Sinclair Methodeoder von der Methode der Eingebetteten Atome hergeleitet sind. # 1999 Acta Metallurgica Inc. Publishedby Elsevier Science Ltd. All rights reserved.

Keywords: Transmission electron microscopy (TEM); Grain boundaries; Theory and modeling of defects

1. INTRODUCTION

The vast majority of engineering materials are poly-crystalline, implying the presence and importance ofgrain boundaries in their processing and perform-

ance. The atomic structure of grain boundaries isimportant for determining such properties as their

migration, sliding, a�nity for segregating species,and resistance to slip transmission. Atomistic simu-lations provide a promising approach to under-

standing and predicting certain aspects of theseproperties. However, atomistic simulations vary in

Acta mater. Vol. 47, Nos 15/16, pp. 3977±3985, 1999# 1999 Acta Metallurgica Inc.

Published by Elsevier Science Ltd. All rights reserved.Printed in Great Britain

1359-6454/99 $20.00+0.00PII: S1359-6454(99)00258-X

{To whom all correspondence should be addressed.

3977

complexity and di�erent materials require di�erent

levels of sophistication to accurately model theirphysics. The face centered cubic (f.c.c.) metals, forexample, are well modeled with potentials devel-

oped with the embedded atom method (EAM) [1, 2]or the Finnis±Sinclair (FS) [3] method and theyhave been used with great success to predict some

unexpected behaviors [4, 5]. These methods havealso been used to develop potentials for body cen-

tered cubic (b.c.c.) metals [6], but here they are lessreliable. For example, they incorrectly predict thesurface reconstruction in W [7] and grain boundary

structure in Nb [8].Interatomic potentials derived within the model

generalized psuedopotential theory (MGPT) [9, 10]

account for many-body angular forces throughexplicit three- and four-body potentials. This contri-

bution to the interatomic interactions is expected tobe important for the central transition b.c.c. metalsdue to the partial ®lling of the d-bands [11]. These

potentials correctly predict the grain boundarystructure in Nb. However, the MGPT potentials aremore computationally intensive (by a factor of

about 40) than the central force potentials men-tioned above and thus it is of interest to determine

whether their use is absolutely necessary in a givensituation.A recent combined theoretical and experimental

investigation has been reported [12] of the S5�310�=�001� and S5 �210�=�001� symmetric tilt grainboundaries (STGBs) in molybdenum. The bound-

aries were simulated using a FS type potential. Aprominent feature of these STGBs arising from the

simulations is the state of rigid body translation ofthe crystals on either side of the boundary, referredto here as ``shifts'' away from the mirror symmetric

structure based on a coincident site lattice (CSL)construction [13]. For these STGBs in Mo, no shiftswere found for the low energy con®guration for

either boundary; both boundaries were predicted tobe mirror symmetric at the atomic scale. Theboundaries were then fabricated and their atomic

structure characterized by HREM. The (210) STGBwas imaged in two projections, one parallel to the

tilt axis and one perpendicular to the tilt axis.Mirror symmetry was found for the (210) STGB.The (310) STGB was only imaged parallel to the tilt

axis and again mirror symmetry was found in thisprojection. No information was obtained for shiftsalong the tilt axis for the (310) STGB from these

experiments. Thus, the experimental results wereinconclusive for the (310) boundary. However,

recent ab initio electronic structure calculations forthe (310) STGB [14] and the (210) STGB [15] inMo indicate a mirror symmetric structure for the

(210) STGB but a shifted structure for the (310)STGB along the tilt axis, at variance with the FSsimulations.

The objective of the present study is to fullycharacterize the three-dimensional structure of the

S5 �310�=�001� STGB in Mo, in particular, tocharacterize the state of rigid body translation

between the crystals on either side of the boundaryand compare with the above-mentioned calcu-lations, as well as MGPT simulations presented

here.

2. ATOMISTIC SIMULATIONS

2.1. The MGPT potential

For an elemental bulk transition metal, the gener-alized pseudopotential theory (GPT) [16] provides a

rigorous real-space expansion of the total energyfor a system of N ions in the form

Etot�R1 . . . RN� � NEvol�O� � 1

2

Xi,j

v2�ij, O�

� 1

6

Xi,j,k

v3�ijk, O� � 1

24

Xi,j,k,l

v4�ijkl, O�

where Ri denotes the position of ion i, O is theatomic volume, and self interaction terms

(i � j � k � l) are excluded from the summations.The leading volume term in the expansion, Evol, aswell as the two-, three-, and four-ion interatomic

potentials, n2, n3, and n4, are volume dependent, butstructure independent quantities and thus transfer-able to all bulk ion con®gurations, including all

structural phases as well as the imperfect bulk solidwith either point or extended defects present. Theangular multi-ion potentials, n3 and n4, re¯ect con-

tributions from partially ®lled d-bands and are gen-erally important for central transition metals. In thefull GPT, however, these potentials are long-ranged,nonanalytic, and multidimensional functions, so

that n3 and n4 cannot be readily tabulated for appli-cation purposes. This has led to the development ofa simpli®ed model GPT or MGPT for b.c.c. tran-

sition metals [9, 10]. In the MGPT, the multi-ionpotentials are systematically approximated by intro-ducing canonical d-bands and other simpli®cations

to achieve short-range, analytic forms, which canthen be applied to both static and dynamic simu-lations. To compensate for the approximationsintroduced, a limited amount of parameterization is

allowed in which the coe�cients of the modeled po-tential contributions are constrained by external ex-perimental or ab initio theoretical data. Useful

MGPT potentials have been so determined for Moover a wide volume range [10] and have recentlybeen applied to the properties of point and line

defects in the bulk [17, 18]. In the present appli-cation, all of the calculations are carried out at a®xed total volume corresponding to the equilibrium

atomic volume O0 � 105:1 a:u: of b.c.c. Mo. Thevolume term Evol is treated as a constant and thepotentials n2, n3, and n4 are applied at the atomicvolume O0.

CAMPBELL et al.: S5 BOUNDARIES3978

2.2. Molecular dynamics implementation

An ideal S5 �310�=�001� STGB was constructedby computer using the rules of the CSL theory. Toaccommodate the interaction range of the MGPT

potentials, the grain boundary plane contained two�1�30� repeat lengths and six [001] repeat lengths.Two [310] repeat lengths were used on each side of

the grain boundary resulting in a simulation cellcontaining N � 960 atoms. Convergence of the cal-culation was checked by doubling the simulation

cell along the grain boundary normal to N � 1920atoms. No additional signi®cant relaxation wasfound and all results reported here employ the N �960 atom cell. Periodic boundary conditions were

applied within the grain boundary plane and freesurface boundaries were applied along the grainboundary normal. However, all atoms within the

MGPT potential interaction range of the free sur-face were ®xed at the bulk lattice positions. These®xed atoms also serve as handles to position the

two grains relative to one another. The system wasbrought to the position of minimum energy using astandard molecular dynamics computer code andthe method of simulated annealing [19].

3. EXPERIMENT

3.1. Grain boundary fabrication

The single crystal of Mo was grown at the

Institute for Solid State Physics in Chernogolovka,Russia, from high purity stock material providedfrom Goodfellow Corp., Cambridge, England. The

Mo was further puri®ed by zone re®ning in highvacuum. The liquid zone was formed by electronbeam heating from an annular electrode and ®vepasses of the liquid zone were taken, with the last

pass starting in a seed crystal oriented for growthalong [310]. The crystal had a diameter of 15 mmand a length of approximately 200 mm.

Two 5-mm slices of the single crystal boule per-pendicular to the growth axis were taken by electricdischarge machining (EDM). The crystal slices were

oriented with Laue backscatter X-ray di�ractionand lapped in specially designed ®xtures to orientthe faces to within 0.18 of (310). The faces werepolished with ¯at polishing techniques to achieve a

¯atness within 100 nm across the 15-mm diameterfaces. Finally, the mutual orientation of the twoslices was set by a 1808 rotation of one crystal with

respect to the other around an axis perpendicular tothe slice face. The process is fully described else-where [20].

Initial attempts to di�usion bond the crystals metwith failure. Simple handling of the bicrystalresulted in brittle fracture at the grain boundary. It

is well known that interstitial impurities, particu-larly oxygen, have a strong embrittling e�ect ongrain boundaries in Mo [21]. Therefore, prior to asubsequent attempt at di�usion bonding, the crys-

tals were given a high temperature treatment of22008C in ultra-high vacuum (UHV, i.e. a pressure

less than 10ÿ7

Pa) for 8 h, which has been shown toreduce the level of interstitial impurities [22]. Thecrystals were then di�usion bonded successfully.

The UHV Di�usion Bonding Machine has beendescribed in detail elsewhere [23]. The Mo crystalswhere sputter cleaned immediately prior to bonding

to remove the oxide layer that forms upon exposureto the atmosphere. Auger spectra taken of the sur-face con®rmed the removal of the oxide. The crys-

tals were bonded at 15008C for 8 h under anapplied load of 1 MPa. The bicrystal was furtherheat treated in a separate UHV apparatus at22008C for 24 h in order to eliminate any residual

porosity remaining at the grain boundary after theinitial di�usion bonding step.

3.2. High resolution electron microscopy

Two cylindrical cores were taken of the bicrystalby wire EDM cutting. The cores were 3 mm in di-

ameter and contained the grain boundary alongtheir axes. The two cylinders were cut with theiraxes along di�erent directions: one along the com-mon [001] direction and the other perpendicular to

the ®rst along the common �1�30� direction. Thecylinders were sliced to give 3 mm disks of 300 mmthickness. The disks were gently lapped to a thick-

ness of 150 mm and 50-mm-deep dimples wereground centered over the grain boundary on bothsides. The ®nal 50 mm was thinned by electropolish-

ing with an electrolyte of 5% sulfuric acid, 35%butoxyethanol, and 60% methanol at ÿ208C. Thedimpling assisted in locating the perforation to

intersect the grain boundary.

Fig. 1. MGPT grain boundary energy plotted vstranslational state along �1�30�.

CAMPBELL et al.: S5 BOUNDARIES 3979

The high resolution imaging was performed on a

JEOL JEM-4000EX operated at 400 keV accelerat-

ing voltage. Images were acquired with an illumina-

tion semi-angle of 0.9 mrad, at a magni®cation of

800 k� , for 1 s exposure time on Kodak SO-163

®lm. For the specimens with [001] foil normal, an

Fig. 2. Structure of the S5 �310�=�001� STGB predicted by MGPT atomistic simulations. The boundaryis viewed (a) along the common [001] directions parallel to the tilt axis and (b) along the common �1�30�

directions perpendicular to the tilt axis.

CAMPBELL et al.: S5 BOUNDARIES3980

objective aperture of radius 6.5/nm was used. For

the specimens with �1�30� foil normal, an objective

aperture of radius 11.8/nm was used, due to the

smaller aperture intersecting the {200} di�raction

disks. The information limit of this microscope is

estimated to be 0.14 nm.

Fig. 2. (continued)

CAMPBELL et al.: S5 BOUNDARIES 3981

High resolution image simulation was performedwith the EMS [24] software package.

4. RESULTS

4.1. Atomistic simulations

It was found that a full relaxation of the freeatomic coordinates starting from the ideal S5�310�=�001� STGB resulted in a relative shift of the

two grains along the [001] direction. This shift isaccommodated by elastic distortion of the planesaway from the grain boundary. To systematically

explore this e�ect, the two grains were shifted, onealong +[001] and the other along ÿ[001], with thegrain boundary plane held ®xed as indicated by the

initial calculation. The resulting atomic coordinateswere relaxed along the grain boundary normal only,that is, a one-dimensional relaxation. The e�ects oftranslational state on the energetics of the grain

boundary were studied by systematically varyingthe relative positions of the adjacent crystals. Theresults for varying the shift along the [001] direction

are shown in Fig. 1. The relative energy monotoni-cally decreases with shift and the unshifted structureis seen to be unstable. The one-dimensional relaxed

structure with minimum energy at a shift of 0.2 a0was used as input for a fully three-dimensional

relaxation. The resulting energy is signi®cantly low-ered suggesting subtle three-dimensional relaxation.

For example, we ®nd the grain boundary plane tobe shifted by about 0.06 a0 along �1�30�.

The structure of the S5 �310�=�001� STGB pre-

dicted from atomistic simulations is shown in Fig.2. Views of the boundary both parallel and perpen-dicular to the tilt axis are shown. When viewed par-

allel to the tilt axis it appears that the boundaryplane is a mirror symmetry plane indicating thesmall shift along the �1�30� direction is imperceptible

to the eye. However, when viewed perpendicular tothe tilt axis, a shift of the crystals is revealed by thedisregistry of the (002) planes across the boundary.This shift breaks the mirror symmetry at the

boundary. The magnitude of the shift is approxi-mately 0.2a0.

4.2. HREM

The grain boundary was observed to form facetswhich were separated by defects that accommodatethe slight misalignment away from the perfect S5orientation of the crystals. The macroscopic mis-orientation of the crystals was measured by Kikuchipattern analysis in thicker regions of the foil. The

deviation away from perfect S5 orientation wasfound to contain a 0.48 tilt rotation and a 0.48 twist

Fig. 3. High resolution transmission electron micrograph of the S5 �310�=�001� symmetric tilt grainboundary in Mo. The viewing direction is along the common [001] direction, parallel to the tilt axis.

CAMPBELL et al.: S5 BOUNDARIES3982

rotation. The defects found in the interface canthen be assumed to contain some dislocation char-

acter to accommodate this misalignment. However,the faceted regions in between the defects appear to

contain a grain boundary of perfect alignment, suit-

able for high resolution microscopy.

The high resolution image acquired parallel tothe tilt axis is shown in Fig. 3. The facets extend

for distances su�cient to escape from the strain®elds of the defects. The grain boundary displays a

plane of mirror symmetry in this projection. To

reveal the shift predicted by the atomistic simu-lations, an image perpendicular to the tilt axis is

required. A high resolution micrograph acquired inthis direction is shown in Fig. 4. Each crystal is

imaged along �1�30�. The two crystal planes with the

largest interplanar spacing containing �1�30� are(002) and (310) at 1.55 and 0.99 AÊ , respectively.

The information limit for this microscope is ap-

proximately 1.4 AÊ and the ``point'' resolution (®rstcrossover of the contrast transfer function at

Scherzer focus) is 1.6 AÊ . Thus, only the (002) set ofatomic planes are imaged as fringes in this image.

Also, this part of the contrast transfer function is

highly damped, leading to the relatively high noiselevel in the image.

The shift of the crystals along [001] is revealed by

sighting down the (002) fringes in the image and

noting their alignment as they cross the grainboundary. The shift is most easily seen in a glancingangle perspective view, such as that shown in Fig.5. The disregistry of the (002) planes as they cross

the grain boundary is evident.

4.3. HREM image simulation

In a qualitative investigation of grain boundary

structure such as this, where the main question tobe answered is whether the grain boundary pos-sesses a [001] shift between the crystals, the mainpurpose of high resolution image simulation is to

con®rm that the image contrast is as expected. It isa way to explore possible imaging artifacts thatwould interfere with the direct interpretation of the

images. Simulated high resolution images are shownin Figs 6(a) and (b) for both orientations in theassumed imaging conditions. In Fig. 6(b) the shift is

clearly evident. An artifact arising from contrastreversal occurring at the interface can be ruled outin the experimental case because of the step change

in thickness required. Likewise, the microscope ima-ging parameters must be the same across theboundary because the crystal orientation is exactlythe same on both sides, meaning that anisotropic

aberrations a�ect both sides equally.

Fig. 4. High resolution transmission electron micrograph of the S5 �310�=�001� symmetric tilt grainboundary in Mo. The viewing direction is along the common �1�30� direction, perpendicular to the tilt

axis.

Fig. 5. Glancing angle perspective view of the same micrograph as shown in Fig. 4 in order to highlightthe shift along the [001] direction at the boundary shown by the disregistry of the (002) planes as they

cross the grain boundary.

CAMPBELL et al.: S5 BOUNDARIES 3983

5. DISCUSSION

The high resolution images of the S5 �310�=�001�STGB in Mo show that the boundary contains arigid body shift of the crystals along [001]. The shift

is evident from a qualitative inspection of Fig. 4,assisted by the glancing angle perspective view

shown in Fig. 5. A quantitative analysis can be per-formed to measure the magnitude of the shift, with

cross-correlation of one grain with respect to the

other being a particularly useful approach [25]. Across-correlation analysis of the image shown in

Fig. 4 gives a relative shift of 0:7820:08 �A , essen-tially 1/4a0.

The cross-correlation analysis also shows the shiftto be constant across the ®eld of view. At both the

extreme right- and left-hand sides of the image, thefringes may appear to favor one side or the other.

However, the quantitative analysis shows that thisvisual perception is incorrect. On the left, it is dueto the wedge of crystalline material ending in amor-

phous material at the edge of the perforation and,on the right, it is due to the facet of perfect bound-ary ending in a grain boundary dislocation.

The magnitude of the measured shift di�ersslightly from the prediction of the present MGPTatomistic simulations, as well as the ab initio results

of ElsaÈ sser et al. [14]. However, the one-dimensionalrelaxations were used to guide the choice of the fullthree-dimensional relaxations in both cases and theprecise position of the energy minimum may di�er.

The results of the atomistic simulations, boththose reported here and those reported in the litera-ture, indicate that the inclusion of angularly depen-

dent interactions is important for proper modelingof the grain boundary structure of b.c.c. metals.The results of simulations using FS potentials [12]

predict no shift, while the MGPT potentials thatinclude these interactions, as well as ab initiomethods [14], predict a shifted structure as the low-

est energy con®guration.

6. CONCLUSIONS

This work has shown that, for certain STGBs inb.c.c. metals with [001] tilt axis, a rigid body shiftcan exist between the crystals, which breaks the

mirror symmetry of the boundary. This shift hasbeen revealed by HREM studies of the S5�310�=�001� STGB in Mo. This shift stands in con-

trast to the structures of the S5 �210�=�001� STGBin Mo [12] and the S5 �310�=�001� STGB in Nb [8],both of which show an unshifted structure when

observed by HREM. These experimental results areconsistent with atomistic simulations that accountfor angularly dependent interatomic interactions in

the potentials.

AcknowledgementsÐWe would like to thank C. ElsaÈ sserand V. VõÂ tek for useful discussions, B. L. Olsen, S. L.Weinland, and R. A. Bliss for their assistance with speci-men preparation, and D. L. Medlin for the use of the4000EX at Sandia National Laboratories, Livermore. Thiswork was performed under the auspices of the UnitedStates Department of Energy and the Lawrence LivermoreNational Laboratory under contract number W-7405-Eng-48.

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