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INCOHERENT Σ3 GRAIN BOUNDARIES IN F.C.C.
METALS : THE INFLUENCE OF INCLINATION ON THE BOUNDARY STRUCTURE AND ENERGY
U. Wolf, P. Gumbsch, H. Ichinose, H. Fischmeister
To cite this version:
U. Wolf, P. Gumbsch, H. Ichinose, H. Fischmeister. INCOHERENT Σ3 GRAIN BOUND- ARIES IN F.C.C. METALS : THE INFLUENCE OF INCLINATION ON THE BOUNDARY STRUCTURE AND ENERGY. Journal de Physique Colloques, 1990, 51 (C1), pp.C1-359-C1-366.
�10.1051/jphyscol:1990157�. �jpa-00230318�
INCOHERENT 23 GRAIN BOUNDARIES IN F.C.C. METALS : THE INFLUENCE OF INCLINATION O N THE BOUNDARY STRUCTURE AND ENERGY
U. WOLF, P. GUMBSCH, H. I C H I N O S E ( ~ ) and H.F. FISCHMEISTER
Maz-Planck-Institut fur Metallforschung, Institut fur Werkstoffwissenschaft. Stuttgart, F.R.G.
Rksumk: La mkthode de "embedded atom" (EAM) est appliquC pour ktudier 1'Cnergie et la structure de diffkrents C3 joints de grains qui sont determinks par leur angle d'inclination
a.
L'Cnergie des joints, yb, varie selona,
par exemple selon l'orientation du plan de joints. La representation graphique pour Cu montre un premier maximum d'knergie pour le joint de macles cohkrent du type (1 11) L des inclinations autour de l'axe [0 1 i]. Un deuxikme minimum d'knergie significant est trouvk B 8' du plan de macles incoherent du type ( 2 11). La localisation d'une configuration d'knergie minimale prBs de (2 11) est en bonne conformitke avec des resultas d'expkriments publiks. Les structures des joints C3 incohkrents autour du second minimum d'knergie montrent des translations rigides qui sont distribuks B travers plusieurs plans de rkseau cristallin, parallels aux interfaces et limitks B un grain.Les structures sont discutCes et comparkes B des observations de macles C3 incohkrent en Au.
Abstract: The "Embedded Atom Method" (EAM) is used to study the energy and structure of different C3 grain boundaries defined by the inclination angle
a.
The grain boundary energy, yb,varies with
a,
i.e. the boundary plane orientation. For copper the yb ( Q ) plot shows a deep energy minimum for the coherent (1 11) twin boundary for inclinations around the [0 1 i] axis. A second significant energy minimum is found 8' away from the incoherent (3 1 1) twin plane. The location of a minimum energy configuration near (2 11) is in good agreement with published experimental results. The structures of the inclined C3 incoherent boundaries of the cusp region show rigid body translations which are distributed over several lattice planes ~ a r a l l e l to the interface and limited to one grain. The structures will be discussed and compared with observations on Au C3 incoherent twins.1 Introduction
Iinomledge of the structure of grain boundaries is necessary in order to understand the properties of poly- crystalline materials, e.g. diffusion, phase transformations, and recrystallization. Observations show that there is a variety of stable orientations between grains. The different types of grain boundaries are usually divided into CSL boundaries, low angle grain boundaries, and random high angle grain boundaries, or more commonly into LLspecial" and "general" grain boundaries. Among the most dominant boundary types in fcc annealed metals are the twinned orientations. Observations on annealed copper show [1,2] that the C3 (2 1 1) twin is inclined relative to a limiting C3 {l 11) twin, i.e., the angle between coherent and incoherent twin deviates from 90" by about 8". Fullman [l] found that there is a clustering of the orientations observed at (1 13) parallel to (3 3 5). Also Omar [2] observed a clustering of all C3 incoherent boundaries around a (3 2 2) olientation of one grain parallel to a (11.4.4) orientation of the other grain on a common [0 1 i] zone axis (in the following all orientations will be denoted as (plane 1)
11
(plane 2), e.g. (3 2 2)11
(11.4.4)).Rigid body translations away from the coincidence configuration generally contribute to atomic relaxation.
'Ipcrrnanent address: University of Tokyo, Institute of Industrial Science, Tokyo, Japan
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990157
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They have been detected in A1-5%Ni C3 incoherent twin boundaries [3] as well as i n pure A1 [4] by analyzing stacking fault fringes. In HREM investigations of Au 2 3 incoherent twins [5] it could be observed that one grain of the C3 incoherent twin shows a rigid body translation relative to the other of a t most 1/2(111) plane spacings depending on t h e distance from a limiting {l l l) coherent twin. This translation is smeared out over a few (3-4) lattice planes parallel t o t h e interface.
In this paper we calculate the grain boundary energy, yb
(a),
of Cu C3 boundaries as a function of the inclination angle, Q, from the C3 (1 1 1 ) coherent twin around the common [0 1 i] axis. T h e structures of the various inclined boundaries are discussed. In the case of Au we simulate the configuartion of a C3 incoherent twin bordering a t one end on a (1 1 1 ) coherent twin i n order to investigate the behaviour of the rigid body translation close t o constrained corners and compare it with the observations i n Au.For the atomistic calculations of Cu and Au bicrystals the "Embedded Atom Method" (EAM) developed l ~ y Daw and Baskes [6] has been applied.
2 Computer Modelling and Results
The geometry of the simulation cell of the Cu and Au bicrystals is shown in fig. 1. Both grains have a common [0 1 i] axis which is the tilt axis of the C3 boundaries. Depending on the orientation relationship of the two grains, the cell contains between 920 and 6192 atoms. In the direction perpendicular to t h e boundary plane periodic boundary conditions were used whereas parallel t o i t free surfaces were considered i n order to admit relaxations of the interface region parallel t o the boundary plane normal. In addition we use t h e constant pressure scheme [7] to account for relaxations within the boundary plane. In all cases we determined tllc structure and energy of the bicrystals by minimizing the total energy (static relaxation) using the conjugated gradient method and EAM potentials. In order to overcome possible metastable energy minima we made a Monte Carlo simulation followed by a static relaxation procedure.
Starting with the C3 (1 1 1 ) coherent twin with inclination angle @ = 0" we produced several C3 orienta- tions by rotating the boundary plane around the [0 l i] axis, ending up with the C3 (2 l l ) incoherent twin at Q) = 90". The yb
(a)
plot (fig. 2) shows a n energy minimum a t @ = 0' and a second cusp a t z 82'. Thc numbers a t each point of fig. 2 correspond to the grain boundaries listed in Table 1.Figure 1: The geometry of the simulation cell of the Cu and Au grain boundaries.
The data points in fig. 3 refer to the relaxed structures; the boundaries are numbered as in Table 1. A remarkable point is that the distributed translations are restricted to only one side of the grain boundary.
Translations of this kind have been observed in HREM investigations on Au [5] (fig. 4). Here the magnitude of the translation varies continuously because the adjacent coherent twin acts on the incoherent twin as a
"rigid barrier". For con~parison with HREM micrographs, a calculation of the structure of a Au C3 (2 1 1 ) incoherent twin has been made. The computer model was adapted t o the geometry of the HREM specimen (fig. 4), by fixing two planes next to the coherent twin boundary. Periodic boundary conditions were used only in the [0 1 i] viewing direction. The result is shown in fig. 5. There are small distortions of the (1 1 1) planes near the coherent twin which successively increase continuously depending on the distance from the coherent twin.
Table 1: List of all configurations computed. @ is the inclination angle, yb and T~ are the grain boundary and surface energy, respectively. The left value in the surface energy column corresponds to the the left grain in the second column, and similarly for the vaIues to the right.
#
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
orientation
(111) [olI]II(i11)
[oli]
(477) [0ii]11(855) [01i]
(122) [Oli]ll(?44) [01i]
(255)[0ii]II(Zii)[o1i]
(144)[oii]11(522)[oii]
(177) [OlT]IJ(3ll) [01i]
(011) [OlI]ll(all) [ O l i ] (i88) [oii]l((TT.2.2)
[oii]
(277) [oli]ll(m.l.i) [oii]
(122) [01I]lJ(T00)
[oii]
(577) [ o ~ i ] J ( ( i i . l . i ) [oif]
( i l l ) [01i]I1(5ii) [oli]
(544) [oli](l(SZZ)
[oii]
(322) [~li]II(fl.a.a) [Olf]
(ii.7.7) [oii]ll(i3.5.5) [oii]
(211) [OlI]II(%I) [Oli]
0.00 13.3 15.8 19.5 25.2 29.5 35.26 40.32 46.69 54.74 62.06 70.53 76.74 81.95 83.28 90.00
X 9 186 215 258 318 358 410 454 494 558 598 606 563 49 1 519 627
1182 1182 1328 1328 1348 1348 1371 1373 1402 1398 1413 1412 1416 1406 1412 1386 1397 1355 1348 1288 1283 1347 1182 1394 1259 1406 1258 1344 1315 1393 1370 1370
-
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Figure 2: Grain boundary energy vs. inclination plot of the Cu C3 grain boundaries. The numbers at each point correspond to the numbers in Table 1
the width of the energy cusp agree with experimental data. Fullman [l] determined the ratio of the energy of the inclined incoherent twin -yt to the grain boundary energy, 7, yt/-l, = 0.80 f 0.015. Taking for the grain boundary energy the value of 625 $[S], this yields .y, = (500 f 10)$;'Omar's [2] measurements give approximately 430
g.
Our computation gives 491 (see Table 1). The computed energy of the coherent twin of about 9g
is much lower than the measured one (45 g )[g]; this is ascribed to the properties of the potentials which do not account for stacking faults. The energies of (1 0 O), (1 1 O), and (1 1 1) surfaces, respectively, agree within less than 1% with other EAM surface energy calculations [10].All grain boundary orientations starting with number 11 of Table 1 show rigid body translations which are continuously distributed over a t most 6 lattice planes parallel t o the interface. The magnitude of the translation vectors varies between approximately 0 and 1.5 (1 11) lattice plane spacings (fig. 3). Such
"bending" of the (1 11) planes has also been observed in pair potential calculations of Cu C3 tilt grain boundaries [11,12]. In these simulations the magnitude of the translation was smaller than in ours because fixed volume was used. These conditions could suppress the occurrence of translational states.
Remarkably, the translations observed are restricted to one grain. The other grain remains rigid without extended relaxations at the interface. Only the incoherent twin is found to make an exception. In this fully symmetric configuration, the translation occurs in one grain or the other depending on an initially introduced disturbance of the mirror symmetry. Thus, the symmetric incoherent twin, which has the highest energy of all orientations studied here, is an unstable grain boundary, and this would seem to be the reason of the tendency to incline the boundary plane.
The observations in Au C3 incoherent twins show that these translations do occur in nature. There, it is obvious that the translation can not be constant in magnitude if some constraint like a coherent twin is situated in the vicinity of the incoherent boundary. The gradual increase of the translation with the distance from the corner agrees with our simulation in Au. More in detail we found that translations between 0 and 1 (1 1 1) lattice plane spacings occur.
The open questions which have still to be discussed are the origin of the rigid body translations in these grain boundaries and the occurence of an inclined incoherent twin boundary. Further investigations on these boundaries will follow this work.
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COLLOQUE DE PHYSIQUE
References
/l/ R.
L.
Fullman: J. Appl. Phys., 22 (1951) 456./2/ R. Omar: Thesis, University of Warwick, Coventry, Great Britain, 1987.
/3/ P.
H.
Pumphrey; T. F. Malis; H. Gleiter: PM. Mag., 34 (1976) 227./4/ R. C. Pond; V. Vitek: Proc. R. Soc. Lond., A 357 (1977) 453.
/ 5 /
H.
Ichinose; Y. Ishida: PM. Mag. A, 52 (1985) 51./6/ M. S. Daw; M. I. Baskes: Phys. Rev. B, 29 (1984) 6443.
/7/ M. Parinello; A. Rahman: J. Appl. Phys., 52 (1981) 7182.
/8/
L. E.
Murr: Interfacial Phenomena in Metals and Alloys, Addison-Wesley, Reading, Mass., 1975./g/ C.
B.
Carter; I. L. F. Ray: PM. Mag., 35 (1977) 189./10/ S. M. Foiles; M. I. Baskes; M. S. Daw: Phys. Rev. B, 33 (1986) 7983.
/11/
A.
P. Sutton; V. Vitek: PM. Trans. R. Soc. Lond. A, 309 (1983) 37./l21
