Calculation of the atomic structure of the ∑ = 13 (θ = 22.6°) [001] twist boundary in gold

Acre mrrull. Vol. 37. X0. IO. pp. 2815~2821. 1989 Printed in Great Britain. All rights resewed

Copyright

WOI-6160 89 $3.00 + 0.00 C 1989 Pergamon Press plc

CALCULATION OF THE ATOMIC STRUCTURE OF THE I: = 13 (0 = 22.6”) [OOl] TWIST BOUNDARY IN GOLD S. M. FOILES Theoretical Division. Sandia National Laboratories. Livermore. CA 945514969. (Be&red

U.S.A.

IO Norember 1988)

Abstract-The structure of the Z = 13 (6 = 22.6’) [OOI] twist boundary in gold is computed using the Embedded Atom Method (EAM). The atomic positions near the boundary are computed and found to be in good agreement with the recent X-ray diffraction results of Fitzsimmons and Sass. The amplitude of the thermal displacements is also computed from the EAM using Monte Carlo simulations and compared with the results for the average Debye-Wailer factor of the boundary obtained by Fitzsimmons. Burke1 and Sass. The vibrational amplitudes of the atoms adjacent to the boundary are found to be larger than in the bulk and the enhancement is greater for vibrations in the plane of the boundary than for vibrations normal to the boundary. These results obtained using the EAM are also compared to results obtained using two diRerent pair interactions. The main difference between the results using the many-body EAM interactions and the pair interactions is that the overall expansion of the boundary is found to bc smaller using the EAM. R&n~&Nous simulons la structure du joint de torsion Z ii: I3 (0 = 22.6’) d-axe [OOI]. dans I’or. i I’aide de la methodc dcs atomcs lies (EAM). Nous calculons des positions atomiqucs au voisinage du joint qui sent en bon accord avec Its resultats rSccnts dc Fitzsimmons et Sass en diffraction dcs rayons X. L’amplitudc dcs dcplacemcnts thcrmiqucs cst tgalcmcnt calculcc i partir dc la mcthodc dcs atomes lies en utilisant dcs simulations de Monte Carlo, et comparce aux rcsultats de Fitzsimmons et Sass sur Ie f’actcur de Dcbye-Wallcr mnycn du joint. L’amplitude dcs vibrations dcs atomcs adjacents au joint est supcrieurc ii ccllc dcs autrcs atomcs, el cellc dill’crencc csl plus importantc pour lcs vibrations parallclcs au plan du joint quc pour Its vibrations pcrpcndiculaircs. Nous comparons nos resultats a ccux qu’on obticnt avcz dcux interactions dc paircs dilTerentr5. La principale dilrcrcnce entre Its rcsultats qui utiliscnt Its interactions EAM i plusicurs corps. et CYUXqui utiliscnt Its interactions de paircs rcsidc darts unc dilation gcncralc du joint infcricurc darts Ic premier cas. %usammenla?rwng-Die Struktur dcr [Olj-Drillkorngrcnzc I: = I3 (I) = 22.6’) in Gold wird mit dcr Mcthodc dcs eingebcttctcn Atoms (EAM) bercxhnct. Die Atompositionen in der Niihe der Korngrenze wcrden bcrrxhnet; sic stimmen gut miut neueren Ergcbnissen aus Riintgcnmcssungcn von Fitzsimmons und Sass i&rein. Dir Amplitude der therm&hen Verschicbungen wird ebcnfalls mit EAM und einer Monte-Carlo-Simulation berechnet; dicse Ergebnisse werdcn mit den von Fitzsimmons, Burke1 und Sass fur den gcmitteltcn Dcbye-Wailer-Faktor crhaltcnen Ergebnissen verglichen. Die Schwingun8samplitudcn der an dcr Korngrenrc Ieigcnden Atome sind grdlkr als in Volumm. der Faktorr its gr6Oer bit Schwingungen in der Ebcne dsr Korngrenze als normal dazu. Diese Ergcbnisse dcr EAM-Rechnungen werden such mit dejenigen von Reichnungen auf der Basis zweicr verschicdcner Paarwechselwirkungcn verglichen. Der Hauptunterschied zwischen den Virlkiirper-EAM-Wcchxlwirkungen und den Paarwechselwirkungen liegt darin. daB die die gemitteltcAusdehnungder Korngrcnze sich bei EAM als kleiner ergibt.

INTRODUCTION

Twist boundaries are good model grain boundaries for understanding grain boundary structure because of their simple gcomctry and the ease with which they can be produced by the Schobcr and Ballufh hot pressing tcchniquc [I]. The structure of these boundaries has been studied experimentally by clcctron microscopy [2] and X-ray diffraction tcchniqucs [3-61 and theoretically by computer simulation [6-IO], Recently, Fitxsimmons and Sass (FS) have dctermincd the atomic structure of the Z = 13 (0 = 22.6’) [OOI] twist boundary in gold using X-ray diffraction techniques [5]. In this paper. the structure of this boundary is computed using the embcddcd atom method (EAM) [I I] and compared to the AM ,7 I&P.

experimentally deterrnincd structure. The agreement of the atomic positions bctwccn theory and cxpcrimcnt is found to be good. This is in contrast to earlier computer simulations of this boundary using pair interaction models where the agreement bctwccn the simulation results and cxpcrimcnt wcrc poor [IZ]. This is also in contrast with the disagreement between the structures computed using pair interaction models and the cxpcrimcntally dctcrmincd structure of the Z = 5 twist boundary in Au (3.4. IO]. This latter disagrcemcnt may bc resolved by the recent combined experimental and theoretical work of Majid [6] on the structure of twist boundaries in Au. In addition lo the structure. the amplitude of the atomic vibrations near the interface are studied via Monte Carlo simulations. Thcx results are compared

2815

3316

FOILES:

STRUCTURE

OF A TWIST BOUNDARY

to the model of the Debye-Wailer factor used by FS in making the experimental analysis of the X-ray diffraction. It is found that the magnitude of the vibrations near the interface are larger than in the bulk crystal. in agreement with the experimental analysis. However, the computer simulation is able to study the vibrations in greater detail. In particular. the simulations show that the increase in the vibrational motion in the plane of the boundary is greater than that normal to the boundary. This differs from the isotropic increase in the amplitudes assumed in the experimental analysis. The first section of the paper will summarize the EAM and describe the calculations. These include the energy minimization calculations to obtain the structure and the Monte Carlo simulations used to obtain the thermal properties. Next the detailed atomic structure of the boundary will be described and compared to the experimentally determined structure and then the vibrational properties will be dcscribcd and rclatcd to the modcling of the Dcbye-Wallcr factor for the interracial atoms. Finally, the results obtained using the many-body EAM interactions will be compared to results obtained using pair intcractions.

CALCULATIONAL

MEI’IIOD

The calculation of the structure and thermal propertics of a gain boundary rcquircs the knowlcdgc of the cncrgy of the system as a function of the atomic coordinates. Because of the large unit cells involved and the large number of configurations that must be considcrcd. a computationally efftcicnt procedure is required to compute the energy. Recently, the embcddcd atom method (EAM) has been proposed by Daw and Baskes [I I] to till this need. This approach requires approximately the same amount of computational etl’ort as modeling the energy by a sum of pair interactions and yet includes important many-body efT&s that arc ignored in the pair interaction trcatments. It has been used successfully in a wide variety of problems including bulk and surface phonon dispersions [I3 141. liquid metal structure (151. surface relaxations and reconstructions [16-l 81, dislocation dynamics [l9). point defect properties [16]. and surface segregation in alloys (201. In the EAM the energy is modeled as having two contributions. The first contribution is an energy which represents the binding of the atom to the local electron density due to the remaining atoms of the system. The second contribution is a pair interaction term reprcscnting electrostatic interactions. In particular, the energy is written as

(1) 1

L

q.r+I

In this expression. F;(p) is the energy to embed an atom into the local electron density p and cp,(R,,) is

IN GOLD

the pair interaction between atoms i and j separated by a distance R,. The electron density at each site is computed from the superposition of atomic electron densities. i.e. P, = c p;(R,). I*#

(2)

In this expression, p;(R) is the electron density of atom j at a distance R from its nucleus. The theoretical motivation of this form has been discussed by Daw [2l] and by Jacobsen et al. [22]. These derivations not only suggest that equations (I) and (2) should provide a reasonable way to model the cohesive energy, they also provide prescriptions for the calculation of the quantities involved from first principles. In the practical application of this method the embedding function. F, and the pair interactions, cp. are determined empirically by fitting to various known properties of the bulk metals. This empirical fitting procedure has been carried out for the f.c.c. metals Cu. Ag. Au, Ni, Pd and Pt by Foilcs. Baskes and Dow (161. In this fitting. the functions wcrc adjusted to rcproducc the zero-tcmpcraturc equation of state. elastic constants and vacancy formation encrgics of the pure metals as well as the dilute heats of solution of the various binary alloys. The details of the functions arc dcscribcd in Ref. [16]. The functions arc used hcrc without any modification. Two types of calculations arc used hcrc to study the structure and thermal propcrtics of the boundary. The first is cncrgy minimization (molczular statics) and the second arc Monte Carlo simulations. In both cases, the gcomctry used for the simulations is that of a bicrystal slab with free surfaces parallel to the boundary. Periodic boundary conditions arc applied in the two directions parallel to the plant of the grain boundary. The rcpcat distance for the periodic boundary conditions, and thereby the area of the boundary, are determined from the lattice constant of the bulk material. In the direction normal to the boundary, the slab is simply terminated by fret surfaces. (The thickness of the slab is increased until the grain boundary has ncgligiblc interaction with the free surfaces.) This geometry simplifies some of the computational problems associated with the volume expansion of the boundary region. In this geometry. the free surfaces of the system can simply shift to adapt to any volume change at the boundary. The energy minimization calculations arc used to determine the zero-temperature structure of the boundary. In these calculations. the coordinates of the atoms are adjusted using a conjugate gradient method [23] to find a local minimum in the potential energy. In this minimization, all of the atomic coordinates arc allowed to vary. Note that since all the atoms are able to displace it is possible for one crystal to move relative to the other crystal. This relative motion can bc either parallel to the boundary or normal to the boundary. The latter is important

FOILES:

STRUCTURE

OF A -hWST

because it allows for the expected expansion at the grain boundary. The difficulty with these calculations is that they find a local potential energy minimum. The conjugate gradient algorithm adjusts the atomic coordinates so as to move downhill on the potential energy surface. Therefore, if there are several local minima in the potential energy surface. the initial values of the atomic positions determine which minimum is found. To attempt to find the global energy minimum. several different initial atomic configurations are used and the energy of the minimized structures are compared. (Frequently, several different initial configurations will result in the same relaxed structure.) The structure with the lowest energy is then taken as the minimum energy structure. The main difference in the study here between the various initial positions is the relative displacement of the two crystals in the plane of the boundary. The second type of calculation used here is Monte Carlo simulation. These simulations determine the equilibrium properties of the system at finite temperature. There are two motivations for this type of simulation. The first is to determine the effect of temperature on the structure and properties of the boundary. The second is to provide an alternative method of finding the optimal structure of the boundary. In essence. the Monte Carlo simulations can be used to anncal the boundary. The procedure for performing Monte Carlo simulations has been described in detail elsewhcrc [24]. In summary, the simulations here follow the Metropolis algorithm. The procedure is as follows. An atom is chosen at random and given a small displacement from its current position. Next, the change in energy, AE. associated with that displacement is computed. If the change in energy is negative, the displacement is retained. If the change in energy is positive, the displacement is retained with a probability given by the Boltzman factor for the energy change, i.e. exp( - AElk, 7). If the change is not retained, the atom is returned to its previous.position. After this cycle is repeated a sufficient number of time, the probability that a configuration will be produced in this way corresponds to thermal equilibrium. Therefore, average properties of the system can be computed simply by averaging over the configurations produced in the simulation. For the simulations performed here, the area of the interface is held Iixcd. (The presence of free surfaces allows the boundary to expand or contract normal to the boundary.) However, because of the thermal expansion of the meatal. the area of the boundary should not be held at the zero-temperature value. To determine the intcrfacial area at tinitc temperature. simulations of the bulk metal are performed at the desired temperature to obtain the lattice constant of the metal at temperature. These bulk simulations are pcrformcd for constant pressure. i.e. the volume is allowed to vary in addition to the atomic positions. (In previous work it was shown that the EAM

BOUNDARY

IN GOLD

2817

predicts thermal expansions in good agreement with experiment [25].) The area of the grain boundary is then determined by this finite temperature lattice constant. As mentioned above, the Monte Carlo simulations can be used to anneal the structure in a search for the optimum grain boundary structure. In a previous study of ZS(?lO)[lOO] tilt boundary. this approach found unexpected structures [26]. To do this, a few of the atomic configurations of the grain boundary produced by the simulations are chosen at random. To eliminate the vibrational part of these structures, the energies of these configurations are minimized as described above. (The size of the system is first scaled to account for the difference between the lattice constant at finite and zero-temperature.) The energy of these minimized configurations are then compared to find the optimal structures. In this present study, the optimum structure obtained in this way is the same as the one found by minimizing assumed structures as discussed above. RFXJLTS The calculations performed here are for the Z I3[OOl] twist boundary in Au. This boundary corrcsponds to a 22.6” rotation around the [OOI] axis. The geometry of this boundary has been described by FS (51and the same designation of the atoms will bc used here to facilitate comparison of the theoretical and experimental results. In the coordinate system used to describe the positions, x corresponds to a (510) axis in each crystal, y corresponds to a (150) axis in each crystal, and r is along the twist axis normal to the boundary, [OOl]. The structure of the unrelaxed coincidence site lattice repeats with a period of mcO along the x and y directions where u, is the lattice constant. The relaxed structure determined for this boundary did not change this periodicity. and the 0

0

X

+

0

+c

0

u

x0

0

+

0

0

0

Ox

0 X0

B

t

+ OX

X

+

X

t 0+

0

0

0

XD

d)o

X

XE

+A

0

X 0

% H

+B XI:

t

0

0

X 0

0

O + %

Fig. I. The unrelaxed Z = I3 twist boundary in Au viewed along the [OOI] twist axis. The different symbols indicate the different atomic planes as indicated on the right. The letters identify the atoms for reference. The horizontal axis corresponds to the x coordinate and the vertical axis corresponds to they coordinate. Atom G is in the plane adjacent to the boundary (the t symbol) and atom H is in the second plane from the boundary (the x symbol).

FO*LES:

2818

STRUCTURE

oF

A - .,._ _ . _._._ 1 WI>1

. ___ __. -__

MUUNIJAKY

IN

GOLD

_

able to the uncertainty of 0.06 A in the experimental [5]. The largest difference, 0.11 A, is for atom H, the coincident atom in the second plane from the boundary. The calculated position places the atom closer to the boundary than the experimental results. This level of agreement between the calculated and experimental positions is much better than that obtained by Bristowe and Sass [12] in previous calculations using pair interactions. The positions obtained in those calculations typically differed from the experimental values by 0.2 A with differences as large as 0.4 A. The Monte Carlo simulations also provide information about the mean square displacements of the atoms near the interface. This information is important because the experimental analysis of the X-ray diffraction requires an estimate of the Debye-Wailer factor for the atoms at the grain boundary. In the experimental analysis by FS. they assumed that all atoms in the two planes adjacent to the boundary had the same Debye-Wailer factor or equivalently the same vibrational amplitude (which was different from the bulk amplitude) and that the vibrations were isotropic. The Debye-Wallcr factor for the interface was determined from the tcmpcrature dcpendcncc of the intensity of the grain boundary reflections by Fitzsimmons er al. [27]. The accuracy of these assumptions about the vibrations at the intcrfacc can be chcckcd by comparing them with the results from the computer simulation studies. The intensity of the scattering for a given scattering vector K is given by the expression [28] tMJk8

-

Two-fold axis

0

Two-fold axis 0

Four-foldaxis

Fig. 2. The symmetry operations of the Z 3: 13 twist boundary in Au. The cell corresponds to that in Fig. I. These symmetry operations can he used to determine the displacementsof the unlabeledatoms in Fig. I.

lowest energy structure corresponds to the coincident site arrangement of the two crystals. Figure I shows the atomic positions of the unrelaxed grain boundary viewed along the twist axis. The figure shows one unit cell of the boundary. As described by FS and indicatcd in Fig. 2. thcrc arc four-fold symmetry axes along : at the center and corners of this figure and two-fold axes along : at the middle of each edge of this cell. Thcrc arc also two-fold axes in the plane of the boundary that run at 45” to the x and y directions and which pass through the midpoint of each cell edge. Because of these symmctrics. it is sufficient to specify the positions of the Iabelcd atoms to charactcrizc the structure. The positions of all other atoms can be obtained using the above symmetry operations. The computed displacements of the atoms from their ideal positions are listed in Tabic I along with the experimental values obtained by FS using X-ray diffraction. Overall the agrecmcnt is quite good. The typical difference between the calculated and experimental positions is about O.OSA which is compar-

W)a:

xe k

-IV,CK,~‘“IK~K,

I. Displacements of

Atom

A.r

A B C D E F G H

the atomic positions away from the ideal coincidcncc silt localions

(A)

0. I7 0.14n f 0.012 -0.06 -0.100 +0.00x 0.21 0.228 + 0.012 -0.03 -0.048 ,t 0.009 -0.01 -0.007 * 0.009 0.07 0.09Of0.011

& 6% -0.04 -0.066 f 0.01 I 0.33 0.296 f: 0.016 -0.OR -o.onx i 0.012 -0.06 -0.106 to.012 0.03 0.04n +0.012 0.03 0.030 f 0.010

(3)

In this expression, the index k indicates atoms and R, is the position of atom k. The quantity W,(K) is the Debye-Wailer factor which is related to the vibrational amplitudes by the expression R’,(K) = 2n2KTB, K

Table

2

(4)

where B, is the matrix of the mean displacements

d- (A) 0.11 0.060 f 0.03x - 0.03 -0.032 ?:0.024 0.19 0.224 f 0.030 0.10 0.109 + 0.028 0.10 0.101 f 0.034 0.10 0.164 f: 0.036 0.01 0.047 f 0.060 -0.03 0.0112f 0.043

The lop values are the present calculakd rcsul~sand the lower values arc the experimental values from Ref. 131.The meaning of Abe atom designations is shown in Fig. I. THc positions of the other atoms a~ the boundary can be determined from these using the symmetry operations shown in Fig. 2. Values not listed arc zero.

Here the brackets denote thermal averages and u, is the difference between the position of the atom and it mean position, i.e. u, = x - (x). In this notation the approximations made in the experimental studies (5,271 are that L$ has the same value for all atoms in the two atomic plants on tither side of the boundary which is diffcrcnt from its value in the bulk, that the off-diagonal elements of Bk vanish, and that the diagonal elements are equal. The matrix Bk has been evaluated for the various atoms in the vicinity of the grain boundary from Monte Carlo simulations performed at room temperature. The results are presented in Table 2. The

FOILES:

STRUCTURE

OF A TWIST

Table 2. Enhancemeot of the vrbrawxul amplitudes of the atoms at UK pain boundary computed from ibc Monte Garb simulation5at room Senqxfature Arom A’ ;: E” E* H



1.76 1.67 1.87 I.15 1.17 1.15 1.36 1.16

(II:>




1.75 1.66 I.83 1.16 I.18 1.14 I.40 1.20

1.30 I .2J 1.31 1.10 1.09 I.12 f.10 1.19

-0.3d

(w:) 0.20 -0.47

-0.46 -0.11 -0.13

The values listed are r&We lo Ihe squared displacements computed for an atom in the bulk. The coordinates and atom designations are as in Fig. I with the atoms in the layer adjacent lo Ihe boundary indicated by an asterisk. ‘the estimated statisMI error is about 0.05 and values not listed are zero to within the statistical uwertain~y.

values in Table 2 are presented as enhancements, i.e. they are the values divided by the value of (u?,) computed in the same manner for atoms far from the boundary. (The squared displacement. (u?), computed for bulk Au at 300 K is 0.~ rf:0.~5 A’. The bulk value obtained by Fitzsimmons el al. 1271 is 0.008 f 0.0005 A* and the literature value 1291 is 0.007 & 0.001 A?.) The statistical uncertainty of the values in Table 2 is about 0.05 and those values not listed are zero to within that uncertainty. The column for (u,u,) is omitted because all of these values were negligible. These results indicate that the vibrational umplitudcs are in fact increased substantially near the grain boundary. By comparison. the exprimentally 1271 estimated enhanccmcnt of the squared displaccmcnts near the boundary is 1.5. The overall magnitude of the computed values are in reasonable acsord with this result, but the simulation results indicate that the assumption of isotropy of the vibrations is not corrt%t. The simulations show that the vibrational motion in the plane of the boundary is approximately isotropic since the compu1c4 values of (uz) 4 (u:) and (u,u,) 5 0. However, the vibrational motion in the plane of the boundary is enhanced more strongly than the motion perpendicular 10 the boundary for atoms in the plane adjacent to the boundary. Furthermore, some of the off-diagonal elements of Bk are significant for elements combining motion in the plane with motion out of the plane of the boundary. The assumption of a uniform value of the vibrational enhancement is also not confirmed by the simulation results. The atom in the coincident site in lhc plane adjacent to the boundary (atom G) has a smaller vibrationat amplitude than the other atoms in that plane. The results also show that the enhancement of the vibrational amplitudes occurs primarily on the plane adjacent to the boundary. The enhancement on the second plant is smaller and the vibrational amplitudcs in the third plant from the boundary are computed to k within a few percent of the bulk values. The current values for the Debye-Wallcr factors at the boundary could be used in a reanalysis of the

BOUNDARY

IN GOLD

2819

experimental diffraction information. Without detailed calculations, one cannot detennine how the details of the Debye-Wallet factor discussed above will change the geometry that is deduced from the experimental scattering data. However, since the overall magnitude of the computed J&bye-Wailer factors are similar to the experimentally assumed value. the change in the experimental geometry should not be substantial. The above results were computed using the EAM to obtain the energetics. The &AM contains many body contributions to the interactions which are absent in the more conventional pair potential calculations of defects structure. it is of interest to know to what extent these results depend on the many-~dy contributions to the energy. To address this. the above calculations were repeated assuming an energy given by the sum of pair potentials. Two different pair potentials were considered. The first is a pair potential deduced from the EAM energies used above. This effective pair interaction approximates 1hc full EAM potential energy for atomic confipuralions close to those of the ideal bulk system. The procedure for extracting the effective pair interaction from equations (I ) and (2) is dcscribcd in Ref. [IS] (xc equation (IO)]. This pair interaction has been used for liquid metals and was shown to repr~u~e the pair correlations ob1ained using the full EAM energy expressions. The difl’crcnce bctwccn results obtained with this eliectivc pair interaction and using 1hc full EAM gives an estimate of the importance of 1he many-body contributions to the energy on the structure and properties of this grain boundary. The second pair potential used for comparison is that due to Baskes and Melius [30]. This empirical pair intcractions was determincd by fitting to similar data as that used in determining the EAM interactions. These pair interactions have been used previously to study defects in metals. For the pair potential calculations, the positions of the atoms near the surfaces were constrained to bulk positions to provide constant volume conditions for the atoms near the boundary so that the volume dependent terms could be ignored. Results from slabs of different thicknesses agreed to within the accuracy quoted. The relaxations computed using the full EAM energy, the effective pair interaction and the Baskes Mclius pair interactions are compared in Table 3. The displacements in the plane of the boundary arc almost independent of the interactions used in the calculations. The difference between the various interactions is reflected in the displacements normal to the boundary. The EAM produces the smallest expansion of the boundary. The results using the effective potential derived from the EAM are similar to those the full EAM except that the atoms arc displaced 0.02 to 0.06 A further from the center of the boundary. This suggests that the main cont~bution of the many-body effects for this boundary is to allow for a somewhat higher overall density at the boundary.

2820

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Tabk 3. Compnrison of the dirpiacerncnts computedruina the BAM (tirrt row), *n &ctive pair iatenrcrionthat approxmata tbc EAM (see Ref. [Id]). and the empirical pair interactionsobtained by Bask= and Mebus(see Ref. 129)).Tbc coordinatesand atom designatioos arc as in Fig. I and values not listedare zero

Atom

hx

(Al

A

0.17 0.18 0.18

3

-0.06 -0.08 -0.08 0.21 0.21 0.22 -0.03 -0.02 -0.02 -0.01 -0.01 -0.02 0.07 0.06 0.07

c

D

D

F

G

H

bv (A) -0.04 -0.04

-0.03 0.33 0.32 0.36 -0.08 - 0.08 -0.08 -0.06 -0.06 -0.05 0.03 0.02 0.03 0.03 0.03 0.04

d: (A) 0.11 0.1s

0.18 -0.03 -0.01 0.02 0.19 0.21 0.27 0.10 0.15 0.23 0.10 0.16 0.24 0.10 0.15 0.26 0.01 0.04 0.10 -0.03 0.03 0.12

The expansion of the boundary region is the largest for the Baskes-Mclius pair interactions which give expansions larger than those obtained in the expcrimcntal analysis. The results obtained with either of thee pair interactions, though, are in better agreement with experiment than the previous pair potential work of Bristowe and Sass [12]. The pair interactions used here are qualitatively different that those used in the earlier study. First, the present interactions are longer ranged including up to third neighbors compared to the previous interaction which included only second neighbors. Also, the general behavior of the interactions are different. The interactions used here are repulsive at short distances and have an attractive well. The interactions used in the previous study are repulsive at short distances, attractive near first neighbor distances but repulsive again around second neighbor distances. These results indicate that pair potentials of the general form considered here provide a better description of the relaxations at a grain boundary. Fitzsimmons PI ul. [27] postulated that the increase in the vibrational amplitudes at the boundary was due to a softening in the interatomic interactions near the Sunday. To test this, the Monte Carlo simulations of the vibrational amplitudes were repeated using the etrective pair interaction discussed above. These results determine the difference between the vibrations for the many-body EAM interactions which include coordination dependence in the interaction strength and an en~ronment inde~n~nr.~ir interaction. The results obtained with the effective pair interaction are very similar to those in Table 2

IN GOLD

except that the enhancement of the amplitudes for the atoms adjacent to the boundary was larger by about 0. I. This shows that the ~b~tional amplitudes at the boundary will be enhanced even in the absence of any change in the interatomic interactions. The enhancement results from the change in the geometry at the boundary. Further, the similarity of the results for the many-body EAM and the effective pair interaction shows that the conb~bution of the environmen~l dependence of the interactions to the enhancement of the vibrational amplitudes is small. SUMMARY

In these calculations the structure of the Z = 13 twist boundary in Au has been computed. The results are in good agreement with the experimental determination of the structure using X-ray diffraction. This agreement supports the reliability of both the theoretical calculations and the experimental procedure. In addition the vibrational amplitudes of the atoms near the boundary have been computed. The magnitudes thus obtained are consistent with the experimental results. However, the simulations are able to determine details that it was not practical to obtain from the experimental analysis and which could be used to refine the dete~ination of the structure from the observed ditfraction. Finally, it was shown that the general results obtained here do not depend crucially on including the many-body effects inherent in the EAM. However, the ability to compute the structure of the boundary using pair interactions to describe the energies depends on the choice of interactions. Acknow&dgme~r~-I

would Iike to thank Professor Stephen L. Sass of Cornell University for providing me with is results prior to publication. Research supported by U.S. Department of Energy, Ofice of Basic Energy Scirnccs, Division of Materials Sciences.

REFERENCFS I. T. Schohcr and R. W. Baluffi. Phil. Mug. 20,511(1969). 2. T. Schohcr and R. W. BaluBi, Phil. &fug. 21, 108 (1970). 3. J. Budai. P. D. Bristowc and S. L. Sass, Actu metcdf.31, 698 (1983). 4. M. R. Fitzsimmons and S. L. Sass, Acre met& 36.3103 (1988). 5. M. R. Fitzsimmons and S. L. Sass. Acru met& 37, 1009 (1989). 6. I. Majid, P. D. Bristowc and R. W. Ballutli. P/ryr. Rev. B. In press. 7, P. D. Bristowe and A. G. Clocker, Phil. Mug. U&487 (1978). 8. G.-J. Wang, A. P. Sutton and V. Vitck. Actu ~t~I. 3% IO93 (1984). 9. P. D. Bristowe and R. W. Baiuffi. 1. Physfque 46 c4-155 (1985). IO. Y. Oh and V. Vitck. Acta merail. 34, 1941 (1986). II. M. S. Daw and M. 1. Baskes. Phys. Rev. futt. SO, 1285 (I 983). Phys. Rev. B 29, 6443 (1984). 12. P. D. Bristowe and S. L. Sass. Actu metuff. 28. 575 (1980). 13. M. S. Daw and R. D. Hatcher. Sotid St. Commzm. 56, 697 (1985).

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