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A systematic study of symmetric tilt-boundaries in hard-sphere f.c.c. crystals
~cra meroll.Vol. 32, No. I, pp. 171-W
1984 Printed in Great Britain. All rights reserved
ooo1-6160/8453.00 + 0.00 Copyright Q 1984Pergmon Press Ltd
A SYSTEMATIC STUDY OF SYMMETRIC TILT-BOUNDARIES IN HARD-SPHERE f.c.c. CRYSTALS M. KOIWA, H. SEYAZAKIt and T. OGURA The Research Institute for Iron, Steel and Other Metals, Tohoku University, Katahira 2chome, Sendai, Japan (Receiued 24 Rprif 1983) Abstract-A
new method is developed for the search of mechanically stable configurations of symmetric tilt boundaries in hard sphere f.c.c. crystals. The problem of finding out relative displacements which minim& the total volume of two crystal blocks forming a boundary, is simplified to a problem of positioning a single atom sphere relative to a block which consists of real and “image” atoms; the lattter are pIaced in such a way to reflect the arrangement in the other crystal block. The method (the image atom method) has been applied to the analysis of symmetric coincidence boundaries with (IOO],[I SO]and [Ill] tilt axes with Cvaluea 3-103. For the [IOO](013)X5 boundary, the procedure of the analysis is described in detail; the derived structures are compared with those by computer simulation. Numerical data are given in tabulated forms for some boundaries. R&am+%Nous pr&entons une nouvelle m&ode de recherche des configurations m6caniquement stables pour ks joints de flexion symttriques dans le mod&Cdes sph&resdures des cristaux c.f.c. Nous ramenons k problhne de la reche&e des dtplaccments relatifs qui minimisent ie volume total des deux blocs cristaliins formant un joint a celui de la mise en place d’une sph&reatomiquc par rapport H un bloc consistant en atomes &Is et “images”; ces demiers sont pla& de man& zi r&l&hir I’armagement dans l’autre bloc cristallin. Nous avons appliquC cettc mCthode(la m&hode des atomes imagea) 4 l’analp des joints de grains symCtriquesde c&cidence dont l’axe de flexion est [ 1001,[ 1IO]et [I 1I] et dont les valeurs de X sont comprises entre 3 et IO3. Nous dtcrivons en d&ail I’analyse pour Ic joint [100](013)ZS; nous comparons les structures ainsi obtenues avec c&es de Ia simulation sur ordinateur. Pour oertains joints de grains, nous donnons des donn&s num&iques sous forme de tabkaux.
Zusammeatrrsuag-Eswird eine neue Methode entwickelt, mit der mechanisch stabik Konfigurationen von symmetrischen Knick-Komgrenzen in einem kfz. Kristall aus harten Kugeht berechnet wenicn k&men. Urn die relativen Verschiebungen, wekhe das Gesamtvolumen der beiden, die Komgmuc bildenden Kristallbiijclre minimaiisieren, N berechnen, wird vereinfachend eine Atomkugei relativ zu einean Block aus reakn und “Bill-Atoms verschoben. Die Bildatome sind so aageordnet,da8 sie die Verteilung im zweiten Block reptisentieren. Mit dieser Methode der Bildatome w&en symmetriache Koinzidenz-korngrenzen mit den Knickachsen [lOO],[I lo] und [ 11l] und den EWerten zwischen 3 und 103analysiert. Als Beispiel wird die Behandlung dcr Komgrenze lOO(Ol3)Z = 5 ausfuhrch beschrieben. SPmtliche so ermittelten Strukturen we&n mit Rachnersimulationen verglichen. Far eiaige KornSrcnxen werden numerische Angaben in Tabellenform gemacht.
1. INTRODUCTION In recent years there has been a strong interest in the properties of grain boundaries, and sever&l attempti
have been made to determine the structural features by means of computer simulation. In such investigations, one assumes an appropriate interatomic potential, and try to find out a minimum energy configuration by relaxation methods (see for example, Refs [I, 21). A general problem in such relaxation studies is that if several local minima exist, a stable configuration, which is found by a single run of simulation, often depends on a starting configuration. Although efforts tPresent address: Mitsubishi Research Institute inc., l-6-1 Otemachi, Chiyoda-ku, Tokyo 100, Japan.
are usually made to examine various starting configurations, one can never be sure if there be some other undiscovered minimum energy configurations. In this situation it is desired to develop a sys~matic method for the search of candidate starting configurations. In the case of f.c.c. crystals, at least, the relative rigid translation of two crystal blocks .fonning a boundary is known to be of great importance in tke reduction of the boundary energy [3J.Therefore, the study of the geometrical structure of boundaries constructed by hard spheres is expected to be useful in the determination of starting con&rations for computer relaxation. Efforts along this line have been made by Frost et al. [4, s]. The present authors have adopted a simitar line, and developed a systematic method for the search of
171
172
KOIWA ef al.: TILT ~UNDARI~
IN HARD-SPHERE f.c.c. CRYSTALS
arrangements; the arrangement in LH seen by atom B, when LH is displaced along the u-direction, is overlapped to the a~angem~t in UH seen by atom A. The atoms thus positioned are depicted by broken lines in Fig. 2; such atoms are referred to as image atoms. Now one considers the positioning of a single atom A relative to the upper half crystal with image atoms. The displacement for the densest configuration can be found easily. The method is referred to as an Image Atom Method (IAM). 2.2. Bosis vectorsfor the description of grain boundaries In order to apply IAM to three-dimensional crystals, we first choose a set of basis vectors which are convenient to define the atom arrangement in a crystal block containing a boundary. A crystal block containing a symmetric coincidence tilt boundary can be regarded as the stacking of certain crystallographic planes parallel to the boundary. It is always possible to find out a set of three basis vectors d,, 6 and d, such that the atom positions in UH (unrotated) can be expressed as
U
Fig. 1. The boundaryseparationvs displacementfor a pair of two alone crystals fR = the radius of atom spheres).For details, see text,
stable iterations which may correspond to local energy minima; the method is referred to as an Image Atom Method (IAM). The aim of this paper is to describe the principle of IAM and some examples of applications. all the rn~~~~y
2. EXPERIMENTAL
2.t. A srr~~h~~~d
~shod
and an iwge atom
method It seemspertinent first to analyze a straightforward
method for finding a dense configuration. Consider two crystal blocks (two-dimensional) placed in a mirror imaga relation as shown in Fig. 1. The problem is to End out the relative displacement u at which the two crystals can be placed most closely (i.e. with the minimum separation, d) without overlapping of atoms. For this purpose, it is convenient to adopt following steps. One shifts the lower half crystal (LH) relative to the upper half (UH) along the udirection. For each displacement is calculated the distance d; that atom A can be placed without overlapping with atoms in UH; in this process one neglects atoms in LH other than atom A. Simiiarfy, 8; is calculated for atom B in UH. By overlapping two curves, separation vs displacement, for atoms A and B, taking larger values for each displacement, one obtains the curve d,, from which the displa~ment for the densest configuration can be found easily. Instead of overlapping the d-curves, the same result can be obtained by overlapping the atom
Id, + m& + nd,
(l,m,n: integers, n 2 0) (1)
where d, and d2 are vectors lying in the crystallographic plane. Similarly, the atom positions in LH (rotated) are given by l’d, + m’dz I- n’d, + T (Z’,m’,n’: integers, n’ Z 0)
(2)
where ‘I’is the.displacement vector, which is 0 for the perfect coincidence boundary, and d, and d, are in a mirror image relation. 2.3. Posiiio~g of image atom For the sake of convenience, a new coordinate system is defined as illustrated in Fig. 3; the x-axis is taken along the vector dl and the z-axis normal to the crystallographic (boundary) plane. The basis vectors in the new coordinate system are denoted with prime: d;, d$, d; and 4. Note that d; and & have the same x- and y-components, and have the same, but opposite in sign, z-components. The displacement vector T’ of LH relative to UH may be written as T’ = (r,,r&*
(3)
Fig. 2. The upper crystal block with superposed “image atoms” (broken lines) and a single atom sphere A which represents the first layer of the lower crystal block. Compare with Fig. I.
KOIWA er al.:
TILT BOUNDARIES
IN HARD-SPHERE
173
T.c.c. CRYSTALS
vectors d; and d;. By virtue of the fact that the displacements T’ and -T’ yield the same relative atom arrangement, however, it is sufficient to examine a half of the unit area. Further reduction in the search area may be possible in some cases, because of the symmetry of the arrangement of real and image atoms, as is shown later. The search for disnlacements or translations which minimize the boundary separation is performed in the following way:
Fig. 3. The coordinate system and the basis vectors d,, d2, and d, for the description of atom arrangements in a crystal containing a boundary. The arrangement in the lower crystal block is expressed by d,, d2, and d,. The vectors d, and d, are in a mirror image relation with respect to the twinning plane.
The densest configuration is represented by the displacement vector having the minimum value of It, I under the restriction of no overlapping of atoms. The positions of “real atoms” in UH are given by XR = id; + md; + nd;, (1,m.n: integers, n r 0).
(4)
The positions of “image atoms”, which reflect the atom arrangement in LH, are X,=I’d;+m’d;-n’d;, (I’,m’,n’: integers, n’ r 0).
(5)
Note that on the first layer real (n = 0) and image (n‘ = 0) atoms occupy the same positions. With such an arrangement of real and image atom spheres, one seeks the position that a sphere of the same size can sit in as close as possible to the above arrangement. The latter sphere is representative of atoms on the first layers of LH and UH, and its position de&s the displacement vector T’. It seems appropriate to note here that the boundary separation is determined by the contact of an atom on the first layer of one crystal block with an atom on the nth layer of the counterpart block. In other words, the boundary separation is never determined by the contact of atoms on the nth and mth layers of the respective blocks with n # 1 and m # 1 (see Appendix). This fact justifies the above-described procedure for the search of the dense configurations. A complicated problem of determining the relative displacement of two crystal blocks is now simplified to a problem of positioning an atom sphere relative to a superposed arrangement of atom spheres. 2.4. Search for dense configurations The displacements corresponding to dense packing should be sought over the unit area spanned by the
(I) A set of three atom spheres (real or image) are chosen among those positioned in and surrounding the search area. All possible sets of three atoms located in close distances from each other should be selected. (2) For each set, the centre position of a sphere of the same diameter circumscribing simultaneously the three spheres is calculated. (3) If the sphere does not overlap with any other spheres, the centre position defines the displacement vector.
3. APPLICATION OF IMAGE ATOM METHOD
By applying IAM, symmetric coincidence tilt boundaries of the f.c.c. crystal with tilt axes, [lOO], [l lo] and [l 111, were examined systematically for Z-values less than 103. Computer programs have been written for the search of all possible displacement vectors. In thii paper various numerical data are given in tabulated forms for boundaries up to E41. Before the explanation of the tables in the next section, various results which can be derived by the use of the tables are demonstrated for a relatively simple boundary (1100](013)CS), the s& feature of which has been studied by several investigators [6-91. The basis vectors of the boundary in the conventional cartesian coordinate system are d, = (2 0 0), d, = (0 3 T), dr=(l
lo), 4-$543)
(6)
where the unit of length is taken as a half of the lattice constant of the f.c.c. crystal. In the new coordinate system explained in 2.3, the vectors are d; =(200), d; = fi(O
lo),
-L(JiG3I), dQz -1-(fi3T) d;=@
1 .
(7)
1 Figure 4 shows the arrangement of atoms indicated by the intersections of spheres cut by the first plane
174
KOIWA
et d.:
TILT BOUN~ARlES
IN HARDSPHERE
kc.
CRYSTALS
cross-section of atom spheres of the upper (solid line) and the lower (broken line) blocks, intersected by the boundary (median) plane between the two crystal blocks. The lower figures visualize the porosity of the respective boundary structures. 4. FORMULAE
AND NUMERICAL
DATA
The numerical data concerning symmetric tilt boundaries with the tilt axes, [IOO], [llO] and [Ill], are summarized in Tables l-3 for 23-41; similar data
for Z-values up to 103are available from the authors. In these tables, the length unit is taken as a half of Fig. 4. The atom arrangement on the first layer of the the lattice constant of the f.c.c. crystal. 1~#]~013~~ boundary (Sequence 5). The circles indicate Table 1 lists the basic data characterizing each the intersections of atom spheres cut by the plane of the boundary. The first column gives a sequential number first layer; the plane ems spheres on the first, second and third layers. Circles with broken lines are image atoms. assigned to each boundary; in the subsequent tables, Small marks (0, A, 0) indicated as A, B and C showthe boundaries are referred to by the sequential number. cmtfe positionsof 3 cimumscribing sphere. The positions The second column classifies the axis of rotation: define the vectors for the relative displacement of the two crystal blocks forming the boundary to attain mechanically stable configurations. The vectors A, B and C give the densest, the second densest and the third densest eonfigurations. The coordinates of the vectors (A, B and C) are
given in Table 3.
of the upper block (f = 0); the largest circles represent the spheres on the first plane (I= 0) and the circles with smaller diameters correspond to spheres on the second (! = 1) and the third (13 2) planes. Circles drawn with broken lines are image atoms placed according to the formula (5). The rectangle formed by the two basis vectors d; and &t is the rcpe& unit of the boundary* The projected centre positions of the contacting spheres, which define the displacement vectors, are also indicated by smaller marks; these are determined by the procedure described in 2.4, There arc three types of mechanically stable configurations (or minima in the boundary separation) and the total eight minima in the unit area. The first, second and third densest positions are indicated by the marks 0, h and Q, respectively. This type of the figure is referred to as the distribution map of the displacement vectors. In order to better visualize the situation, the d, YS displacement curves such as in Fig. 1 are drawn in Fig. 5; the separation of the boundary, d,, is calculated along the lines
1 [loo], 2 [llO], 3 [Ill]. The third, fourth and fifth columns give X-values, the rotation angle, and the twinning plane, respectively. The sixth column (LA) is the number of layers in one period in the direction normal to the twinning plane {the z-axis); this number is the same as or twice of the &value. The seventh column (NT) is the number of types of the displacement vectors which give mechanically stable positioning of the two (upper and lower) crystal blocks; the compotrents of each vector arc given in Table 3. The eigbtb column (D,,& is the planar spacing, and the ninth column (R,) is the repeat distance along the y-direction (normal to the rotation axis, x). The tenth column gives the area of the repeat unit of the boundary. The last column (V,) is the excess volume per atom-arca (or the number of atoms “missing” per atom-area in the boundary} for the densest configuration; the quantity has been defined by Frost et ul. [4]. Table 2 lists the three basis vectors, d,, dz and d, in the original coordinate system. There are a number
Fig. 5. The boundary separation vs displacement curves along the lines indicated in the upper figure.
KOIWA
*
et ai.:
o
OA
TILT A
BOUNDARIES
IN HARD-SPHERE
h-x.
CRYSTALS
f75
of ways of choosing a set of basis vectors. In this paper, the following rules are adopted for the choice.
06.
(I) The vectors d, and d, lie in the twinning plane, (HKL).
(2) The set, d,, dz and d, forms a right-handed system: (d, x d&d, >O. (3) The vector d, and the plane normal, nHKL,point to the same side (the upper crystal) of the plane d, *nHKL> 0. (4) The vector d, is taken parallel to the rotation
axis. With these genera1 r&s, there still remains room for the choice of d, and d,; we choose the shortest one each among an infinite number of candidates. By examining the three w of the rotation axes, [lOO], [ 1lo] and [ 11 I], separately, the axpressions, useful for practical application arc derived as follows. f 100]-axis d, =[200], d, =
[ILR] { [OLR]
oddK-L even K-L
63)
[ 1IO]-axis dl = [l IO], d, =
[1L2fl odd H+L { [L’L’+ LH] even H+ L L’= -[(L - W2M
(9)
(11 II-axis d, = [2 2 21, d,=[M+KM-HM]
Fig. 6. The side view of the [1~](013)~5 boumiaries constructed with the displacement vazton: (a) A, (b) I3 and (c) C. Circles with solid and broken lines arc atoms belonging to the upper and the lower crystals, respectively.
HZ-K2-L2
Mm=
’ H2+Kzi-L2
2HK 2LH
t[alG represents the maximum integer which is not greater than e. $For Sequence 6, d, is determined so as to make d, ad, > 0 rather than to minimize ld,(.
M=Nfor
H-K-3Nor
M=N+l
forH-K=3N$l.
3N+2
(10)
All the data shown in the Table, except for Sequence 6, are derived according to the above rules.% The vector d,, which defines atom positions in the lower crystal block together with d, and d,, is obtamed by rotating the vector d, by 180” with respect to the plane normal, [HKL], and reversing the direction:
2HK K2-L2-#
(11)
2KL
For practical applications, it is convenient to CXpress basis vectors in a new coordinate system as defined below:
KOIWA
176
et al.:
TILT
BOUNDARIES
IN HARD-SPHERE
x-direction: the direction of the vector d,, i.e. parallel to the rotation axis. y-direction: normal to both the x- and zdirections. z-direction: normal to the twinning plane (HKL).
f.c.c. CRYSTALS
(13)
where F = l/,/m.
When the rotation axis and the twinning plane
(HKL) are specified, the directions of the x-, y- and
z-directions are defined by: Rotation axis
[I x001 [I 101 [11 11
001 u 101 111 11 [l
The basis vectors in the new coordinate system, d; , d; and d; are obtained by applying the transformation matrices, R,OO,RtlO,or R,,, , depending upon the axis of rotation.
where
F, = lI,/%
F2 = I/,/w,
Fj = I/,/m.
Table I. Symmetric tilt grain boundaries in the f.c.c. lattice No.
: 3 4 : 7 : IO II I2 I3 14 15 16 I7 18 I9 20
AX
II
e
2 2 3 1
3 3 3 5 5 7 9 9 II II I3 I3 I3 I7 I7 17 I7 19 19 19 21 25 25 27 27 29 29 31 33 33 33 33 37 37 37 39 41 41 41 41
70.53 70.53 60.00 36.87 36.87 38.21 38.94 38.94 50.48 50.48 22.62 22.62 27.80 28.07 28.07 86.63 86.63 26.53 26.53 46.83 21.79 16.26 16.26 31.59 31.59 43.60 43.60 17.90 58.99 58.99 20.05 20.05 18.92 18.92 50.57 32.20 12.68 12.68 55.88 55.88
: 2 2 2 2 I : I : 2 2 2 :
: 23 24 :: 27 28 29 30 31 32 33 34 3s 36 37 38
I : 2 I : 2 2 2 2
I
: 3 I I :
Plpne
LANT
&XL
3 6 6
1 3 3
t! I4 I8
: 4 4
1: 22
42 6
;: 26 34 34
: 7 3 6
z 38
: 7
:: 42 50 50 27 54 58
9 3 9 5 5 2 8 5
:
I:
: 66 66 74 74 74 78 82 ::
1: 8 7 5 9 I3 I3 6 IO 7
82
8
AX-rotution axis; I: [IOO]. 2: [IIO]. 3: (III]. LA-number of layers in one period (I direction). NT-number of types of displacwnent vcclors. II,,,-interplanar spacing. R,-repeat distance (normal IO : and rotation axis). Area-areu of repeat unil. VEy-excess volume for the densest conliguration (Type I).
I.1547 0.4082
0.4082 0.4472 0.3162 0.2673 0.2357 0.3333 0.6030 0.2132 0.2774 O.l%l O.l%l 0.2425 0.1715 0.2425 0.1715 0.1622 0.4588 0.1622 0.1543 0.2Otnl 0.1414 0.3849 0.1361 0.1857 0.1313 0.1270 0.1741 0.1231 0.1231 0.1741 0.1644 0.1162 0.1162 0.1132 0. I562 0.1104 0.1104 0.1562
R, 2.4495 3.4641 14142 4.4721 3.1623 6.4807 :!! 416904 6.6332 7.21 I I ::z 8.2462 5.8310 5.8310 8.2462 8.7178 6.1644 IO.6771 3.7417 10.000 7.0711 7.3485 10.3923 lo.no3 7.6158 13.6382 8.1240 11.4891 11.4891 8.1240 12.1655 8.6023 14.8997 5.0990 12.8062 9.0554 12.8062 9.0554
Arm 1.7321 4.8990 4.8990 4.472 1 6.3246 7.4833 8.4853 !E 9:3808 7.2111 10.1980 10.1980 8.2462 11.6619 8.2462 Il.6619 12.3288 4.3589 12.3288 12.9615 10.000 14.1421 5.1962 14.6969 10.7703 15.2315 15.7480 II.4891 16.2481 16.2481 I I.4891 12.1655 17.2047 17.2047 17.6635 12.8062 IS.1108 18.1108 12.8062
VEZ 0.0000
0.2092 0.2092 0.3221 0.3194 0.3753 0.3742 0.2688 0.1260 0.3413 0.4013 0.4176 0.4425 0.3850 0.4285 0.3046 0.3481 0.4504 0.2407 0.4292 0.4756 0.4518 0.4490 0.3055 0.4362 0.4422 0.4640 0.4948 0.4145 0.3921 0.4813 0.4171 0.4591 0.4706 0.4439 0.4791 0.4859 0.4869 0.4682 0.3738
KOIWA Table
2. Basis
NO
AX
z
I 2 3 4 5 6 7 8 9 IO II I2 I3 I4 15 16 I7 I8 I9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
2 2 3
3 3 3 5 5 7 Y Y
I I 3 2 2 2 2
3
I I 2 2 2 2 3 3
I I 2 2
I 1 3 2 2 2 2
I I I I 2 7_
her grain
II IO)
(
(110) (22 2) 1200)
( ( ( i ( ( (-I ( (-2 ( ( ( ( ( ( (-4 ( (0 ( ( ( ( ( (-2 ( ( ( ( (-4 ( (-I ( ( ( ( ( ( ( (-3
(222)
(1 10) (1 10) (1 IO) (1 IO) (200) (2 0 0) (2 2 2) (200) (200) (110) (IlO) (1 IO) (1 10) (2 2 2) (2 2 2) (200) (200) (1 IO) (I 10) (200) (200) (222) (110) (110) (110) (110) (200) (200) (222) (222) (200) (200) (1 10) (1 IO)
BOUNDARIES
0 2-2 I-I I 0 3 4-4
IN HARD-SPHERE
boundaries
4
coo;
:: 37 37 39 41 41 41 41
3 3
vectors
4
II II 13 13 13 17 17 17 I7 19 19 19 21 25 25 27 27 29 29 31 33 33 33
I I
er 01.: TILT
4 I
I) 2) 0)
I-21 3 -Ii 0 I) 2) I 4) 2-l I) 2 6) I 2-3) 0 5-l) 3-l 0) I 4-l) 0 3 -5) 3-3 4) 4 6) 6-6 2) l 3) 2-3-l) 3-2-l) I 3 -4) 0 7-l) 3-2 I) 2 10) I 5 -2) 0 3-7) 3-3-2) 5-5 4) 4 IO) 8-8 2) I 8) I 6-l) 0 5-7) 3-4-l) 4-3 -1) I 4 -5) 0 9 -I) 8-8 6) 3 8)
I ( 0 (-I I-I i I ( O-I (-I ( I ( l-l ( I ( 0 ( l ( 2 ( 1 (-1-I ( 0 (0 (-2 ( I (-I ( O-I-I) ( 0 ( l ( I-I (
0 I) I -I) 0 -I) 0 I) I oi -I) 2-l) O-I) 0) O-I) I-I) l 0) l I) I 0) 2) l-l) I 1) 3-l) O-I) I 0) l-l) I
0) 0)
( 0 I 3) (-I -2 1) ( I l-2) ( O-I-I) ( 2-l 1) ( I O-I) (-3 4-l) ( I O-3) ( I 1 0) (-1-2 3) (-I I 0) (-1 2 I) ( 0 l-l) ( 1 1 0) ( 2-l I) ( I 0 -1)
T.c.c.
177
CRYSTALS
In the last column, the positive and negative signs correspond to the vectors d; and d;, respectively. Table 3 lists. for X3-21. components of displacement vectors, T’, (in the new coordinate system), which give mechanically stable hard-sphere contact configurations of two crystal blocks: the lower half
crystal should be displaced relative to the upper half by such an amount defined by the displacement vector. The vectors are arranged in the increasing order of the boundary separation or the absolute magnitude of the z-component; type A corresponds to the densest configuration, type B the next densest and so on. As stated before, there are several displacement vectors which yield equivalent configurations; only one of each type with the smallest IT’ 1 is listed. Note that for a given displacement vector T T:=
+T’+fd;+md;;
f,m=O,
+l,
+2 ,...
(16)
is also a displacement vector of the same type. For boundaries with the rotation axes, [IOO] and [l lo], additional vectors of the same type exist by virtue of the presence of the symmetry planes normal to the rotation axis. The atom arrangement of a crystal block containing the boundary is given by: upper block: Id; + md; + nd; I,m,n, integers, n 2 0 (17)
lower block: I’d; + m’d; + n’d; + T I’,m ‘,n ‘, integers, n ’ 2 0. 1
FI h,,
Fl
FI
F,(H + 2K) - F,(2H + K)
=
F,H
i
F,(H - K) - F,(H + K) i (15)
F3K
where F2 = 1/,/6(H2 + HK + K’),
F, = II&
F3 = l/J2( HZ + HK + K*). The vector 4 is obtained by changing the sign of the z-component of the vector d;. By applying the appropriate transformation matrix to the basis vectors d, given in Table 2, one obtains the vectors d; in the new coordinate system; some examples are given below for several boundaries, No. I-No.6. Sequence No.
AM.,211 --L
d;
It should be added here that for T’ = 0, the first layers of the upper and the lower blocks completely overlap; the layer is doubly occupied by spheres belonging to the two blocks. In comparing the displacement vectors reported by other investigators with the present ones, one must remember this fact. Figure 7 shows some further examples of the distribution maps of the displacement vectors constructed from the information given in the Tables, for the boundaries, the structure of which has been studied by previous investigators: (a) (b) (c) (d)
Sequence Sequence Sequence Sequence
4 6 9 11
[100](02l)ZS [111](123)211 [110](113)211 [ 100](032)213
Refs Ref. Refs Refs
d;, d;
d;
1
(GOO)
UIJWPO)
(IIJi
2
($00)
(0 2*
(lI~‘lJ5
f J5)
3
(2J500)
(O$O)
(2/Y/5 I/$
f Jm
4
(200)
(1fiO)
(12/d
5
(2 0 0)
(OflO)
U3lfi
6
(2J5 0 0)
(40
0)
J%
0)
(4lJ3
l/fi
* 218)
f l/J% * VJlo) SI@
* l/J%.
[6, 7, 81 [lo] [ll, 91 [8, 61.
178
KOIWA
e! a/.:
TILT BOUNDARIES IN HARD-SPHERE f.c.c. CRYSTALS Table 3. Displacement vectors
No. 1 2 3 4 5
6
A
A A A A A
7
D A D
a
A
9 10
II 12 13
A
0.3440
A D A
A D A :
16
A
D
19 20
l.Wlf
D
A A D
18
0.4605
0.2727 2.7942 0.7071 0. 0.3903 0.7071
14 IS
17
O.fO?l 0. 1.7321
0.1940 0. 0.9303 0.6417 1. 1.4334 I.1547 I.4891 0. 0.1340 0.7854
0.7071 0.
A
0.
:
0.7071
A
it
:
0.
A
0. 1.53% 1.0300 0.7698
A
D G
0.4082-1.t547 1.7321-0.6719 -0.6719 0. I .0297-0.8530 1.581ko.7187 0.1424-0.7402 I .2533-0.9708 I. -0.7071 1.7961-0.8649 2.1213-0.6720 0.7071-l. 1.1408-0.7617 3.3166-0.6433 t.ao9t-o.a528 1.~~.78~ 2.5495-0.7223 0.1%1-0.9806 0.2175-0.7536 0.2265-0.784s 1.9378-0.9359 1.2127-0.7276 2.9155-0.7114 1.251~.82~ 2.PlfS-0.6264 1.0290-0.9701 4.1231-0.6101 2.9913-0.7432 0.9701-1.0290 2.4374-0.7297 2.~7~.7~8 4.3589-0.8509 1.2329-0.7620 1.3424-0.7030 1.779!&0.7315 0.8117-0.8652
B ::
0. I.1547
B B
kl277 0.9017
B
0.7071 0.7698
0.769a-O.9526 0.7071-0.9526
E
I. 1.1547
0.3 162-O.po87
J.0801-0.7776
0.707 1
1.3224-0.7533
c
0.
3.
B
0.
1.7641-0.7423
c
0.
1.0999-0.8889
B B E ::
0.7071 0.7071 0. 00:7226
0.9239-WMO 2.7526-0.7500 l.ooso-o.9949 0,887~.a3# 1.1767-o.7845
C F
0.7071 0.7071
:
;:77o4
2.2111-0.7817 0.6030-1.0660 0.5547-o.8322 1.9642-0.7937
c
0.6773 1.7240
0.9624-0.7843 1.47209.8974
0.4472-0.8944
0.8474 1.3ooa
0.8339-0.7658 0.8286-0.7937
8”
0: 05609
E
0.6294
! E”
0.7071 0.7071 0.
H
0.7071
;
II:7071
1.8106-0.8067 2.669!W.?263 0.9~.~78 2.2293-0.7276 0.6288-1.0510 3.62lO-O.7101 2.182a-o.8575 0.5659-1.0862 2.3099-0.7380 0.9177-o.8111
B B E H
0. 1.5259 1.0755 2.5852
1.1896-0.7647 1.3219-O.7043 1.3313-O.7532 ~1.3M8-O.9421
5. I. Comparison with boundury structures by computer ~~uiut~n
It is interesting to compare the atom configurations of grain boundaries obtained by computer simulation and those predicted by the present analysis with IAM. For the [1001(013) boundary, Hashimoto ef al. [l’2] have found two types of structures, Type I and Type 2, the former one is similar to those reported by Weins et al. [6] and Smith el a/. [8]. The structures shown in Fig. 6(a) and (b) are very similar to Type 1 and that in Fig. 6(c) to Type 2 (compare with Figs 3 and 5 in Ref. [12]). The compa~~n can be made more clearly by plotting the displacement vector corresponding to each simulated structure in the map, as shown in Fig. 8. The vectors are located very close to those derived in the present analysis.
scheme.
is translated
1.1772-0.7732
B E
5. DISCUSSION
present
c C
In these figures the projection of the vector d, is shown by the broken line. Note that the vector I$ is not generally orthogonal to the vector dt; the area of the rectangles in the figures is two or three times of that of the repeat unit spanned by d, and d2.
Vhe x, v and z terminology
1.1547-0.8165 -0.8165 0.
to conform
to the
F
-0.7917
;:
O.2425-0.9701 2.1671-0.8266 0.5145-0*8575 1.8293-O.8085
0.0900 0.
3.3485-0.7246 1.7786-o.9147
0.1834 0.7071
2.7284-0.7532 1.7816-o.8314
0.7071
1.6244 1.1547
0.8113-0.9177 0.0829-0.7la1 0.0937-0.811 I
2.8587
0.8246-0.9765
;:
: F I
1.t5435.80~8
5.2. The determination of initial con&rations
The procedure for determination of initial previous investigations (1) Weins et al. [6],
the rigid translation or the configurations adopted by the are: (WGC)?
l‘. . . the crystals were translated from their coincidence configurations in the ydirection and in the x-direction in increments of l/20 of the period in the y-direction, and l/10 of the period in the x-direction. At each of the y- and x-translations the energy minimum by relaxation in the z-direction was determined using an increment in z of 0.1 A which corresponds to a step of approximately l/5 of the minimum step size in the y-direction.. . .‘I’ (2) Smith et al. [8], (SVP) l‘. . . The procedure adopted was to begin with the coincidence structure and allow this to relax.. . . Next, the first plane of atoms parallel to the boundary was removed from the upper crystal.. . . In this way, up to 6 planes were removed su&essivdy from the upper and the lower grains before relaxation was carried out.. . .” (3) Hashimoto et al. [12], (HIYD) 6, . . , The upper grain was moved relative to the lower grain by small increments parallel to the boundary plane without rotation and maintaining the atoms on the original lattice sites of their respective
KUIWA
et aI_: TILT BOUNDARfES
d, =‘(200),
IN HARD-SPHERE
d,=(k!),
d, =(-ioi
kc.
CRYSTALS
1
(4
d, =(2221,
d,=f301),
d,-f0ii)
d, =(IT0t
d, =( 110) , (c)
d, -(200),
cl,‘tt231,
d3=fOlTi
Fig. 7. The distribution maps of the displacement vectors for several boundaries. Note that the recta&s arc not the repeat unit of respective boundaries in these examples. The displacement vectors giving rn~~n~~lly stable ~n~gu~tions were sought over a half or a one-fourth of the repeat unit spanned by the vectors d, and d,. The vector d, is appropriately displaced so as to point to inside the rectangle. (a) Sequence [4] [1OO](OZl)ZS,(b) Sequence 6 [111](123)1;7 (c) Sequence 9 [I lO](l13)211. (d) Sequence 11 [lOO](O32)r;13.
179
180
KOlWA
et al.:
TILT ~UNDARIES
IN HARD-SPHERE
f.c.c. CRYSTALS
plane. Figure 9 indicates the displacements corresponding to the removal of 0, I, 2,. . . ,Q layers, for the 25 boundary. For each boundary, there exists a number of layers, NT, the removal of which results in apparently different configurations. For the present CS boundary, the number is 10 (see Table 1). AS seen in Fig. 9, the successive removal of layers scans the whole area first sparsely and then minutely; the procedure seems to be very reasonable. Nevertheless, one examines equivalent configurations if the method is mechanically applied. in the case of Fig. 9, the following pairs are equivalent: (0 8), (1 7), (2 6), and (3 5). Note that the marked points in the figure regularly Fig. 8. Comparison of displacnnent vectors determined by align on certain lines. Since the minimum points may previous investigators for the [100](013)25 boundary. The locate at places distant from those lines as seen in Fig. locations marked as W and S are those determined by Weins 8, there might be a possibility of missing the lowest er al. [6] and Smith et ul. [S], respectively. H, and H, refer energy configuration by starting only from those to “type I” and “type 2” in Ref. 1121.The marks,0, A and 0, representthe displacement vectors determined in the configurations produced by the removal of layers. presentstudy. 5.3, Excess volume of grain boundaries crystals. The increments spectively, l/10 and l/30
of translations
were, re-
of the periods in the X- and y-directions.. . . The same process was continued for five steps. In each succeeding step the increments of translation were made smaller than that of the previous step. . . . in order to determine the minimum energy configuration, the above calculation process was started from various structures made by removing or adding a ‘few atomic layers from or to the Z5 coincidence lattice site ~lations~p . . .”
The difference between the volume of a bicrystal and that of a perfect single crystal is the definition of the “excess volume” associated with the grain boundary. Frost et al. [4,5] have caIculated the excess volume, Vsx of the densest configurations for 51 symmetric coincidence tilt boundaries. We have extended the calculations and obtained the values for total 98 boundaries (CfC103). The values are plotted as a function of the area of the repeat unit, as shown in Fig. 10. The excess volume increases with the increase in the area of the repeat unit, and seems to have a station value less than 0.5, which co&rrns the trend observed by Frost et. al. [5j. It is added here that we have compared our results with those given in the “catalogue” by Frost et al. 1131,for all the non-faceted densest boundaries with C-values less than 45; an excellent general agreement is found as it should be. One exception is the
In the light of the result of the present investigation, some remarks are made on the efficiency of the above described procedures for the determination of starting configurations, The procedure of drawing a grid over a certain area and calculating the energy for each grid point, adog ted by WGC!and HIYD, seems to be a natural one; the finer the mesh size, the better one has the opportunity of encountering a minimum energy point. In adopting this way of the search, it is worthwhile to remember that only a half (the [ill] rotation axis) or one fourth (the [lOOI and [lIOJ rotation axes) of the area of the repeat unit is to be examined. At each of the y- and n-translations, WGC tried to find the energy minimum by allowing the relaxation in the z-direction, while HIYD seem to have calculated simply the energy for a fixed z-translation. The present hard sphere analysis is similar, in some sense, to the WGC analysis, and is expected to yield results with a good correspondence to those by WGC but not to those by HIYD. SVP produced different initial configurations by Fig. 9. Displacement vectors ~uiva~ent to the removal of successiveremoval of layers followed by the shift of layers for the constriction of initial structures for computer the relevant crystal by one interplaner spacing, which is equivalent to certain displacements in the x-y
simulation. The numerals in the figure correspond to the number of layers removed.
KOIWA er al.:
TILT BOUNDARIES
IN HARD-SPHERE
f.c.c. CRYSTALS
181
0.60
0 30 t 0
0.20 0
0.10 ~
0.01 + 0
2
’
’
’
4
6
B
’
f
’
’
’
’
’
’
’
t
1
10 12 14 16 I8 20 22 24 26 28 30
Area
of repeat
unit
Fig. 10. Excess volume as a function of the area of the repeat unit of boundaries.
[l 10](115)~27; our v, is 0.3055 while theirs is 0.3053. As to the sizes of substitutional or interstitial holes given in their catalogue, minor differences with our results were noticed only for the [100](032), [I 11](145), and [100](043) boundaries. 6, CONCLUSIONS AND SUMMARY
A systematic method is developed for the study of the symmetric tilt boundary models between hard sphere f.c.c. crystals. The analysis begins by finding a proper set of basis vectors in describing the atom arrangement in crystals containing a boundary. The problem of finding mechanically stable and/or dense configurations of two crystal blocks can be simplified by introducing the concept of “image” atoms. The atom arrangement in one block is overlapped with that in the other, so that the problem is reduced to the positioning of a single atom sphere relative to one crystal block containing both real and image atom spheres. The new method, referred to as the image atom method (IAM), allows a systematic, yet economical and thorough search for the mechanically stable configurations. The method has been applied to 98 symmetric coincidence tilt boundaries with [loo], [llO] and [ll l] tilt axes, and numerical results are given in tabulated forms for some boundaries. There are generally several stabIe configurations for a given boundary. Some boundaries have as many as 20 different stable configurations or the displacement vectors. The derived structures were compared with those obtained by computer simulation with a qualitatively good agreement. In the light of the present analysis, some remarks have been made on the validity of the procedure, adopted by previous investigators, for the choice of initial configurations for computer simulation. It is concluded that a careful analysis of the geometry of
the hard sphere model of boundaries provides useful information for more advanced investigations by using realistic interatomic potentials.
~~~~e~~e~~f~-~e authors are greatly indebtecl to Dr S. Ishioka for useful advice and eniightening discussion. REFERENCES
Harrison, G. A. Bruggeman and G. H. Bishop, &a& Boundary Structure and Properties (edited by G. A. Chadwick and D. A. Smith), p. 45. Academic Press, New York (1976). 2. V. Vitck, A. P. Sutton, D. A. Smith and R. C. Pond, I. R. J.
GrainBounaiuyStructureand Kinetics,p. I 15. Am. Sot. Metals, Metals Park, OH (1980). 3. V. Vitek, R. C. Pond and D. A. Smith, Computer Simulationfor Material Application, N&ear Metalkg-y, Vol. $0, p. 265. Plenum Press, New York (1976). 4. H. I. Frost, M. F. Ashbv and F. Snaenen. . . . Scrivta . metall. 14, iOSl (1980). * 5. H. J. Frost, F. Spaepen and M. F. Ashby, Scripro metall. 16, 1165 (1982). 6. M. J. Weins, H. Gleiter and B. Chalmers, f. u&. Phys. 4% 2639 (I 971). 7. 0. Hasson, J.-Y. Boos, I. Herbeuval, ,M. Biscondi and C. Goux, Surf. Sci. 31, I15 (1972). 8. D. A. Smith, V. Vitek and R. C. Pond, Acta metall.25, 475 (1977). 9. N. Hashimoto, Y. Ishida, R. Yamamoto and M. Doy-
ama, Acta met&f. 29, 617 (1981). IO. G. Hasson, M. Biscondi, P. Lag&e, J. Levy and C. Goux, The Nature and Bet&or of Grain 3ounalaries (editui by H. Hu), p. 3. Plenum Press, New York (1972). 1 I. R. C. Pond, D. A. Smith and V. Vitek, Acta metoll.27, 235 (1979). 12. M. Hashimoto, Y. Ishida, R. Yamamoto and M. Doyama, J. Phys. F, Metal Phys. 10, 1109 (1980). 13. H. J. Frost, M. F. Ashby and F. Spaepen A ~ruIogue of [ ftX& f [lo], and [tl I] Symmetric Tilt Boundariesin Face-Centered Cubic Hard Sphere Crystals. Harvard Univ. Press (1982). APPENDIX In the s~rne~c
tilt grain boundaries of the hard-sphere model, spheres on the first layer are always in touch with spheres belonging to the counterpart block, thus determining the boundary separation. This can be proved by showing that the following situation cannot happen. “A sphere P in the mth layer of the upper block is in contact with a sphere Q in the nth layer of the lower block, where m z n > I.” For simplicity’s sake, we adopt a new coordinate system explained in Section 2.3, and further take the x-axis along
KOIWA er al.:
182
TILT BOUNDARIES
the d, direction. Then the vectors d;, d; and the displacement vector T’ for the relevant boundary are written as
IN HARD-SPHERE
Similarly, in the lower block a sphere Q’ exists at -(n - 1)d; from Q, i.e. in the first layer Q’ = Q - (n - 1)d;
d; = (x0, yo. ~a), d; = (xgl YO.-20) T’ = (XT, YT. -IT),
IT >
= (x2 - (n - t)-%, Y2 - (n - IlYo. -+I.
0.
The square of the distance p’e’
The coordinates of spheres P and Q are YI, z,)=(x,,
P=(x,,
Q = (x,, YZ, 23 = h
f.c.c. CRYSTALS
is
(p’ - Q)2 = (x, - x2)2
YI. (m - lb,)
+ (_Y,-yd2 + [(m - n)za + zJ2.
~2, -61 - llz,, - zT).
(A2)
By comparing with equation (Al)
Since the two spheres are in contact, one has
(I” - Q’)2 < 4r,. (P-Q)2=4r: or
(Al)
(x, --x*)2 + (_I$- YJ’ + [(m + n - 2)za + ZJ = 4G I where r, is the sphere radius. In the upper block, there exists a sphere P’ at -(a - l)d; from P, i.e. in the (m - n + 1)th layer, P=P-(n-l)d; = (x, -(n
- 1)x,, YI -(n
- l)Yo, (m - nk).
Evidently, this cannot happen, since spheres cannot overlap each other in the hard-sphere model. In order to satisfy the relation, (P’-Q)224rf or [(m - n)za + zrl’ 2 [(m + n - 2)~s + 2$, n must be unity. This proves the statement at the beginning of Appendix.
1984 Printed in Great Britain. All rights reserved
ooo1-6160/8453.00 + 0.00 Copyright Q 1984Pergmon Press Ltd
A SYSTEMATIC STUDY OF SYMMETRIC TILT-BOUNDARIES IN HARD-SPHERE f.c.c. CRYSTALS M. KOIWA, H. SEYAZAKIt and T. OGURA The Research Institute for Iron, Steel and Other Metals, Tohoku University, Katahira 2chome, Sendai, Japan (Receiued 24 Rprif 1983) Abstract-A
new method is developed for the search of mechanically stable configurations of symmetric tilt boundaries in hard sphere f.c.c. crystals. The problem of finding out relative displacements which minim& the total volume of two crystal blocks forming a boundary, is simplified to a problem of positioning a single atom sphere relative to a block which consists of real and “image” atoms; the lattter are pIaced in such a way to reflect the arrangement in the other crystal block. The method (the image atom method) has been applied to the analysis of symmetric coincidence boundaries with (IOO],[I SO]and [Ill] tilt axes with Cvaluea 3-103. For the [IOO](013)X5 boundary, the procedure of the analysis is described in detail; the derived structures are compared with those by computer simulation. Numerical data are given in tabulated forms for some boundaries. R&am+%Nous pr&entons une nouvelle m&ode de recherche des configurations m6caniquement stables pour ks joints de flexion symttriques dans le mod&Cdes sph&resdures des cristaux c.f.c. Nous ramenons k problhne de la reche&e des dtplaccments relatifs qui minimisent ie volume total des deux blocs cristaliins formant un joint a celui de la mise en place d’une sph&reatomiquc par rapport H un bloc consistant en atomes &Is et “images”; ces demiers sont pla& de man& zi r&l&hir I’armagement dans l’autre bloc cristallin. Nous avons appliquC cettc mCthode(la m&hode des atomes imagea) 4 l’analp des joints de grains symCtriquesde c&cidence dont l’axe de flexion est [ 1001,[ 1IO]et [I 1I] et dont les valeurs de X sont comprises entre 3 et IO3. Nous dtcrivons en d&ail I’analyse pour Ic joint [100](013)ZS; nous comparons les structures ainsi obtenues avec c&es de Ia simulation sur ordinateur. Pour oertains joints de grains, nous donnons des donn&s num&iques sous forme de tabkaux.
Zusammeatrrsuag-Eswird eine neue Methode entwickelt, mit der mechanisch stabik Konfigurationen von symmetrischen Knick-Komgrenzen in einem kfz. Kristall aus harten Kugeht berechnet wenicn k&men. Urn die relativen Verschiebungen, wekhe das Gesamtvolumen der beiden, die Komgmuc bildenden Kristallbiijclre minimaiisieren, N berechnen, wird vereinfachend eine Atomkugei relativ zu einean Block aus reakn und “Bill-Atoms verschoben. Die Bildatome sind so aageordnet,da8 sie die Verteilung im zweiten Block reptisentieren. Mit dieser Methode der Bildatome w&en symmetriache Koinzidenz-korngrenzen mit den Knickachsen [lOO],[I lo] und [ 11l] und den EWerten zwischen 3 und 103analysiert. Als Beispiel wird die Behandlung dcr Komgrenze lOO(Ol3)Z = 5 ausfuhrch beschrieben. SPmtliche so ermittelten Strukturen we&n mit Rachnersimulationen verglichen. Far eiaige KornSrcnxen werden numerische Angaben in Tabellenform gemacht.
1. INTRODUCTION In recent years there has been a strong interest in the properties of grain boundaries, and sever&l attempti
have been made to determine the structural features by means of computer simulation. In such investigations, one assumes an appropriate interatomic potential, and try to find out a minimum energy configuration by relaxation methods (see for example, Refs [I, 21). A general problem in such relaxation studies is that if several local minima exist, a stable configuration, which is found by a single run of simulation, often depends on a starting configuration. Although efforts tPresent address: Mitsubishi Research Institute inc., l-6-1 Otemachi, Chiyoda-ku, Tokyo 100, Japan.
are usually made to examine various starting configurations, one can never be sure if there be some other undiscovered minimum energy configurations. In this situation it is desired to develop a sys~matic method for the search of candidate starting configurations. In the case of f.c.c. crystals, at least, the relative rigid translation of two crystal blocks .fonning a boundary is known to be of great importance in tke reduction of the boundary energy [3J.Therefore, the study of the geometrical structure of boundaries constructed by hard spheres is expected to be useful in the determination of starting con&rations for computer relaxation. Efforts along this line have been made by Frost et al. [4, s]. The present authors have adopted a simitar line, and developed a systematic method for the search of
171
172
KOIWA ef al.: TILT ~UNDARI~
IN HARD-SPHERE f.c.c. CRYSTALS
arrangements; the arrangement in LH seen by atom B, when LH is displaced along the u-direction, is overlapped to the a~angem~t in UH seen by atom A. The atoms thus positioned are depicted by broken lines in Fig. 2; such atoms are referred to as image atoms. Now one considers the positioning of a single atom A relative to the upper half crystal with image atoms. The displacement for the densest configuration can be found easily. The method is referred to as an Image Atom Method (IAM). 2.2. Bosis vectorsfor the description of grain boundaries In order to apply IAM to three-dimensional crystals, we first choose a set of basis vectors which are convenient to define the atom arrangement in a crystal block containing a boundary. A crystal block containing a symmetric coincidence tilt boundary can be regarded as the stacking of certain crystallographic planes parallel to the boundary. It is always possible to find out a set of three basis vectors d,, 6 and d, such that the atom positions in UH (unrotated) can be expressed as
U
Fig. 1. The boundaryseparationvs displacementfor a pair of two alone crystals fR = the radius of atom spheres).For details, see text,
stable iterations which may correspond to local energy minima; the method is referred to as an Image Atom Method (IAM). The aim of this paper is to describe the principle of IAM and some examples of applications. all the rn~~~~y
2. EXPERIMENTAL
2.t. A srr~~h~~~d
~shod
and an iwge atom
method It seemspertinent first to analyze a straightforward
method for finding a dense configuration. Consider two crystal blocks (two-dimensional) placed in a mirror imaga relation as shown in Fig. 1. The problem is to End out the relative displacement u at which the two crystals can be placed most closely (i.e. with the minimum separation, d) without overlapping of atoms. For this purpose, it is convenient to adopt following steps. One shifts the lower half crystal (LH) relative to the upper half (UH) along the udirection. For each displacement is calculated the distance d; that atom A can be placed without overlapping with atoms in UH; in this process one neglects atoms in LH other than atom A. Simiiarfy, 8; is calculated for atom B in UH. By overlapping two curves, separation vs displacement, for atoms A and B, taking larger values for each displacement, one obtains the curve d,, from which the displa~ment for the densest configuration can be found easily. Instead of overlapping the d-curves, the same result can be obtained by overlapping the atom
Id, + m& + nd,
(l,m,n: integers, n 2 0) (1)
where d, and d2 are vectors lying in the crystallographic plane. Similarly, the atom positions in LH (rotated) are given by l’d, + m’dz I- n’d, + T (Z’,m’,n’: integers, n’ Z 0)
(2)
where ‘I’is the.displacement vector, which is 0 for the perfect coincidence boundary, and d, and d, are in a mirror image relation. 2.3. Posiiio~g of image atom For the sake of convenience, a new coordinate system is defined as illustrated in Fig. 3; the x-axis is taken along the vector dl and the z-axis normal to the crystallographic (boundary) plane. The basis vectors in the new coordinate system are denoted with prime: d;, d$, d; and 4. Note that d; and & have the same x- and y-components, and have the same, but opposite in sign, z-components. The displacement vector T’ of LH relative to UH may be written as T’ = (r,,r&*
(3)
Fig. 2. The upper crystal block with superposed “image atoms” (broken lines) and a single atom sphere A which represents the first layer of the lower crystal block. Compare with Fig. I.
KOIWA er al.:
TILT BOUNDARIES
IN HARD-SPHERE
173
T.c.c. CRYSTALS
vectors d; and d;. By virtue of the fact that the displacements T’ and -T’ yield the same relative atom arrangement, however, it is sufficient to examine a half of the unit area. Further reduction in the search area may be possible in some cases, because of the symmetry of the arrangement of real and image atoms, as is shown later. The search for disnlacements or translations which minimize the boundary separation is performed in the following way:
Fig. 3. The coordinate system and the basis vectors d,, d2, and d, for the description of atom arrangements in a crystal containing a boundary. The arrangement in the lower crystal block is expressed by d,, d2, and d,. The vectors d, and d, are in a mirror image relation with respect to the twinning plane.
The densest configuration is represented by the displacement vector having the minimum value of It, I under the restriction of no overlapping of atoms. The positions of “real atoms” in UH are given by XR = id; + md; + nd;, (1,m.n: integers, n r 0).
(4)
The positions of “image atoms”, which reflect the atom arrangement in LH, are X,=I’d;+m’d;-n’d;, (I’,m’,n’: integers, n’ r 0).
(5)
Note that on the first layer real (n = 0) and image (n‘ = 0) atoms occupy the same positions. With such an arrangement of real and image atom spheres, one seeks the position that a sphere of the same size can sit in as close as possible to the above arrangement. The latter sphere is representative of atoms on the first layers of LH and UH, and its position de&s the displacement vector T’. It seems appropriate to note here that the boundary separation is determined by the contact of an atom on the first layer of one crystal block with an atom on the nth layer of the counterpart block. In other words, the boundary separation is never determined by the contact of atoms on the nth and mth layers of the respective blocks with n # 1 and m # 1 (see Appendix). This fact justifies the above-described procedure for the search of the dense configurations. A complicated problem of determining the relative displacement of two crystal blocks is now simplified to a problem of positioning an atom sphere relative to a superposed arrangement of atom spheres. 2.4. Search for dense configurations The displacements corresponding to dense packing should be sought over the unit area spanned by the
(I) A set of three atom spheres (real or image) are chosen among those positioned in and surrounding the search area. All possible sets of three atoms located in close distances from each other should be selected. (2) For each set, the centre position of a sphere of the same diameter circumscribing simultaneously the three spheres is calculated. (3) If the sphere does not overlap with any other spheres, the centre position defines the displacement vector.
3. APPLICATION OF IMAGE ATOM METHOD
By applying IAM, symmetric coincidence tilt boundaries of the f.c.c. crystal with tilt axes, [lOO], [l lo] and [l 111, were examined systematically for Z-values less than 103. Computer programs have been written for the search of all possible displacement vectors. In thii paper various numerical data are given in tabulated forms for boundaries up to E41. Before the explanation of the tables in the next section, various results which can be derived by the use of the tables are demonstrated for a relatively simple boundary (1100](013)CS), the s& feature of which has been studied by several investigators [6-91. The basis vectors of the boundary in the conventional cartesian coordinate system are d, = (2 0 0), d, = (0 3 T), dr=(l
lo), 4-$543)
(6)
where the unit of length is taken as a half of the lattice constant of the f.c.c. crystal. In the new coordinate system explained in 2.3, the vectors are d; =(200), d; = fi(O
lo),
-L(JiG3I), dQz -1-(fi3T) d;=@
1 .
(7)
1 Figure 4 shows the arrangement of atoms indicated by the intersections of spheres cut by the first plane
174
KOIWA
et d.:
TILT BOUN~ARlES
IN HARDSPHERE
kc.
CRYSTALS
cross-section of atom spheres of the upper (solid line) and the lower (broken line) blocks, intersected by the boundary (median) plane between the two crystal blocks. The lower figures visualize the porosity of the respective boundary structures. 4. FORMULAE
AND NUMERICAL
DATA
The numerical data concerning symmetric tilt boundaries with the tilt axes, [IOO], [llO] and [Ill], are summarized in Tables l-3 for 23-41; similar data
for Z-values up to 103are available from the authors. In these tables, the length unit is taken as a half of Fig. 4. The atom arrangement on the first layer of the the lattice constant of the f.c.c. crystal. 1~#]~013~~ boundary (Sequence 5). The circles indicate Table 1 lists the basic data characterizing each the intersections of atom spheres cut by the plane of the boundary. The first column gives a sequential number first layer; the plane ems spheres on the first, second and third layers. Circles with broken lines are image atoms. assigned to each boundary; in the subsequent tables, Small marks (0, A, 0) indicated as A, B and C showthe boundaries are referred to by the sequential number. cmtfe positionsof 3 cimumscribing sphere. The positions The second column classifies the axis of rotation: define the vectors for the relative displacement of the two crystal blocks forming the boundary to attain mechanically stable configurations. The vectors A, B and C give the densest, the second densest and the third densest eonfigurations. The coordinates of the vectors (A, B and C) are
given in Table 3.
of the upper block (f = 0); the largest circles represent the spheres on the first plane (I= 0) and the circles with smaller diameters correspond to spheres on the second (! = 1) and the third (13 2) planes. Circles drawn with broken lines are image atoms placed according to the formula (5). The rectangle formed by the two basis vectors d; and &t is the rcpe& unit of the boundary* The projected centre positions of the contacting spheres, which define the displacement vectors, are also indicated by smaller marks; these are determined by the procedure described in 2.4, There arc three types of mechanically stable configurations (or minima in the boundary separation) and the total eight minima in the unit area. The first, second and third densest positions are indicated by the marks 0, h and Q, respectively. This type of the figure is referred to as the distribution map of the displacement vectors. In order to better visualize the situation, the d, YS displacement curves such as in Fig. 1 are drawn in Fig. 5; the separation of the boundary, d,, is calculated along the lines
1 [loo], 2 [llO], 3 [Ill]. The third, fourth and fifth columns give X-values, the rotation angle, and the twinning plane, respectively. The sixth column (LA) is the number of layers in one period in the direction normal to the twinning plane {the z-axis); this number is the same as or twice of the &value. The seventh column (NT) is the number of types of the displacement vectors which give mechanically stable positioning of the two (upper and lower) crystal blocks; the compotrents of each vector arc given in Table 3. The eigbtb column (D,,& is the planar spacing, and the ninth column (R,) is the repeat distance along the y-direction (normal to the rotation axis, x). The tenth column gives the area of the repeat unit of the boundary. The last column (V,) is the excess volume per atom-arca (or the number of atoms “missing” per atom-area in the boundary} for the densest configuration; the quantity has been defined by Frost et ul. [4]. Table 2 lists the three basis vectors, d,, dz and d, in the original coordinate system. There are a number
Fig. 5. The boundary separation vs displacement curves along the lines indicated in the upper figure.
KOIWA
*
et ai.:
o
OA
TILT A
BOUNDARIES
IN HARD-SPHERE
h-x.
CRYSTALS
f75
of ways of choosing a set of basis vectors. In this paper, the following rules are adopted for the choice.
06.
(I) The vectors d, and d, lie in the twinning plane, (HKL).
(2) The set, d,, dz and d, forms a right-handed system: (d, x d&d, >O. (3) The vector d, and the plane normal, nHKL,point to the same side (the upper crystal) of the plane d, *nHKL> 0. (4) The vector d, is taken parallel to the rotation
axis. With these genera1 r&s, there still remains room for the choice of d, and d,; we choose the shortest one each among an infinite number of candidates. By examining the three w of the rotation axes, [lOO], [ 1lo] and [ 11 I], separately, the axpressions, useful for practical application arc derived as follows. f 100]-axis d, =[200], d, =
[ILR] { [OLR]
oddK-L even K-L
63)
[ 1IO]-axis dl = [l IO], d, =
[1L2fl odd H+L { [L’L’+ LH] even H+ L L’= -[(L - W2M
(9)
(11 II-axis d, = [2 2 21, d,=[M+KM-HM]
Fig. 6. The side view of the [1~](013)~5 boumiaries constructed with the displacement vazton: (a) A, (b) I3 and (c) C. Circles with solid and broken lines arc atoms belonging to the upper and the lower crystals, respectively.
HZ-K2-L2
Mm=
’ H2+Kzi-L2
2HK 2LH
t[alG represents the maximum integer which is not greater than e. $For Sequence 6, d, is determined so as to make d, ad, > 0 rather than to minimize ld,(.
M=Nfor
H-K-3Nor
M=N+l
forH-K=3N$l.
3N+2
(10)
All the data shown in the Table, except for Sequence 6, are derived according to the above rules.% The vector d,, which defines atom positions in the lower crystal block together with d, and d,, is obtamed by rotating the vector d, by 180” with respect to the plane normal, [HKL], and reversing the direction:
2HK K2-L2-#
(11)
2KL
For practical applications, it is convenient to CXpress basis vectors in a new coordinate system as defined below:
KOIWA
176
et al.:
TILT
BOUNDARIES
IN HARD-SPHERE
x-direction: the direction of the vector d,, i.e. parallel to the rotation axis. y-direction: normal to both the x- and zdirections. z-direction: normal to the twinning plane (HKL).
f.c.c. CRYSTALS
(13)
where F = l/,/m.
When the rotation axis and the twinning plane
(HKL) are specified, the directions of the x-, y- and
z-directions are defined by: Rotation axis
[I x001 [I 101 [11 11
001 u 101 111 11 [l
The basis vectors in the new coordinate system, d; , d; and d; are obtained by applying the transformation matrices, R,OO,RtlO,or R,,, , depending upon the axis of rotation.
where
F, = lI,/%
F2 = I/,/w,
Fj = I/,/m.
Table I. Symmetric tilt grain boundaries in the f.c.c. lattice No.
: 3 4 : 7 : IO II I2 I3 14 15 16 I7 18 I9 20
AX
II
e
2 2 3 1
3 3 3 5 5 7 9 9 II II I3 I3 I3 I7 I7 17 I7 19 19 19 21 25 25 27 27 29 29 31 33 33 33 33 37 37 37 39 41 41 41 41
70.53 70.53 60.00 36.87 36.87 38.21 38.94 38.94 50.48 50.48 22.62 22.62 27.80 28.07 28.07 86.63 86.63 26.53 26.53 46.83 21.79 16.26 16.26 31.59 31.59 43.60 43.60 17.90 58.99 58.99 20.05 20.05 18.92 18.92 50.57 32.20 12.68 12.68 55.88 55.88
: 2 2 2 2 I : I : 2 2 2 :
: 23 24 :: 27 28 29 30 31 32 33 34 3s 36 37 38
I : 2 I : 2 2 2 2
I
: 3 I I :
Plpne
LANT
&XL
3 6 6
1 3 3
t! I4 I8
: 4 4
1: 22
42 6
;: 26 34 34
: 7 3 6
z 38
: 7
:: 42 50 50 27 54 58
9 3 9 5 5 2 8 5
:
I:
: 66 66 74 74 74 78 82 ::
1: 8 7 5 9 I3 I3 6 IO 7
82
8
AX-rotution axis; I: [IOO]. 2: [IIO]. 3: (III]. LA-number of layers in one period (I direction). NT-number of types of displacwnent vcclors. II,,,-interplanar spacing. R,-repeat distance (normal IO : and rotation axis). Area-areu of repeat unil. VEy-excess volume for the densest conliguration (Type I).
I.1547 0.4082
0.4082 0.4472 0.3162 0.2673 0.2357 0.3333 0.6030 0.2132 0.2774 O.l%l O.l%l 0.2425 0.1715 0.2425 0.1715 0.1622 0.4588 0.1622 0.1543 0.2Otnl 0.1414 0.3849 0.1361 0.1857 0.1313 0.1270 0.1741 0.1231 0.1231 0.1741 0.1644 0.1162 0.1162 0.1132 0. I562 0.1104 0.1104 0.1562
R, 2.4495 3.4641 14142 4.4721 3.1623 6.4807 :!! 416904 6.6332 7.21 I I ::z 8.2462 5.8310 5.8310 8.2462 8.7178 6.1644 IO.6771 3.7417 10.000 7.0711 7.3485 10.3923 lo.no3 7.6158 13.6382 8.1240 11.4891 11.4891 8.1240 12.1655 8.6023 14.8997 5.0990 12.8062 9.0554 12.8062 9.0554
Arm 1.7321 4.8990 4.8990 4.472 1 6.3246 7.4833 8.4853 !E 9:3808 7.2111 10.1980 10.1980 8.2462 11.6619 8.2462 Il.6619 12.3288 4.3589 12.3288 12.9615 10.000 14.1421 5.1962 14.6969 10.7703 15.2315 15.7480 II.4891 16.2481 16.2481 I I.4891 12.1655 17.2047 17.2047 17.6635 12.8062 IS.1108 18.1108 12.8062
VEZ 0.0000
0.2092 0.2092 0.3221 0.3194 0.3753 0.3742 0.2688 0.1260 0.3413 0.4013 0.4176 0.4425 0.3850 0.4285 0.3046 0.3481 0.4504 0.2407 0.4292 0.4756 0.4518 0.4490 0.3055 0.4362 0.4422 0.4640 0.4948 0.4145 0.3921 0.4813 0.4171 0.4591 0.4706 0.4439 0.4791 0.4859 0.4869 0.4682 0.3738
KOIWA Table
2. Basis
NO
AX
z
I 2 3 4 5 6 7 8 9 IO II I2 I3 I4 15 16 I7 I8 I9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
2 2 3
3 3 3 5 5 7 Y Y
I I 3 2 2 2 2
3
I I 2 2 2 2 3 3
I I 2 2
I 1 3 2 2 2 2
I I I I 2 7_
her grain
II IO)
(
(110) (22 2) 1200)
( ( ( i ( ( (-I ( (-2 ( ( ( ( ( ( (-4 ( (0 ( ( ( ( ( (-2 ( ( ( ( (-4 ( (-I ( ( ( ( ( ( ( (-3
(222)
(1 10) (1 10) (1 IO) (1 IO) (200) (2 0 0) (2 2 2) (200) (200) (110) (IlO) (1 IO) (1 10) (2 2 2) (2 2 2) (200) (200) (1 IO) (I 10) (200) (200) (222) (110) (110) (110) (110) (200) (200) (222) (222) (200) (200) (1 10) (1 IO)
BOUNDARIES
0 2-2 I-I I 0 3 4-4
IN HARD-SPHERE
boundaries
4
coo;
:: 37 37 39 41 41 41 41
3 3
vectors
4
II II 13 13 13 17 17 17 I7 19 19 19 21 25 25 27 27 29 29 31 33 33 33
I I
er 01.: TILT
4 I
I) 2) 0)
I-21 3 -Ii 0 I) 2) I 4) 2-l I) 2 6) I 2-3) 0 5-l) 3-l 0) I 4-l) 0 3 -5) 3-3 4) 4 6) 6-6 2) l 3) 2-3-l) 3-2-l) I 3 -4) 0 7-l) 3-2 I) 2 10) I 5 -2) 0 3-7) 3-3-2) 5-5 4) 4 IO) 8-8 2) I 8) I 6-l) 0 5-7) 3-4-l) 4-3 -1) I 4 -5) 0 9 -I) 8-8 6) 3 8)
I ( 0 (-I I-I i I ( O-I (-I ( I ( l-l ( I ( 0 ( l ( 2 ( 1 (-1-I ( 0 (0 (-2 ( I (-I ( O-I-I) ( 0 ( l ( I-I (
0 I) I -I) 0 -I) 0 I) I oi -I) 2-l) O-I) 0) O-I) I-I) l 0) l I) I 0) 2) l-l) I 1) 3-l) O-I) I 0) l-l) I
0) 0)
( 0 I 3) (-I -2 1) ( I l-2) ( O-I-I) ( 2-l 1) ( I O-I) (-3 4-l) ( I O-3) ( I 1 0) (-1-2 3) (-I I 0) (-1 2 I) ( 0 l-l) ( 1 1 0) ( 2-l I) ( I 0 -1)
T.c.c.
177
CRYSTALS
In the last column, the positive and negative signs correspond to the vectors d; and d;, respectively. Table 3 lists. for X3-21. components of displacement vectors, T’, (in the new coordinate system), which give mechanically stable hard-sphere contact configurations of two crystal blocks: the lower half
crystal should be displaced relative to the upper half by such an amount defined by the displacement vector. The vectors are arranged in the increasing order of the boundary separation or the absolute magnitude of the z-component; type A corresponds to the densest configuration, type B the next densest and so on. As stated before, there are several displacement vectors which yield equivalent configurations; only one of each type with the smallest IT’ 1 is listed. Note that for a given displacement vector T T:=
+T’+fd;+md;;
f,m=O,
+l,
+2 ,...
(16)
is also a displacement vector of the same type. For boundaries with the rotation axes, [IOO] and [l lo], additional vectors of the same type exist by virtue of the presence of the symmetry planes normal to the rotation axis. The atom arrangement of a crystal block containing the boundary is given by: upper block: Id; + md; + nd; I,m,n, integers, n 2 0 (17)
lower block: I’d; + m’d; + n’d; + T I’,m ‘,n ‘, integers, n ’ 2 0. 1
FI h,,
Fl
FI
F,(H + 2K) - F,(2H + K)
=
F,H
i
F,(H - K) - F,(H + K) i (15)
F3K
where F2 = 1/,/6(H2 + HK + K’),
F, = II&
F3 = l/J2( HZ + HK + K*). The vector 4 is obtained by changing the sign of the z-component of the vector d;. By applying the appropriate transformation matrix to the basis vectors d, given in Table 2, one obtains the vectors d; in the new coordinate system; some examples are given below for several boundaries, No. I-No.6. Sequence No.
AM.,211 --L
d;
It should be added here that for T’ = 0, the first layers of the upper and the lower blocks completely overlap; the layer is doubly occupied by spheres belonging to the two blocks. In comparing the displacement vectors reported by other investigators with the present ones, one must remember this fact. Figure 7 shows some further examples of the distribution maps of the displacement vectors constructed from the information given in the Tables, for the boundaries, the structure of which has been studied by previous investigators: (a) (b) (c) (d)
Sequence Sequence Sequence Sequence
4 6 9 11
[100](02l)ZS [111](123)211 [110](113)211 [ 100](032)213
Refs Ref. Refs Refs
d;, d;
d;
1
(GOO)
UIJWPO)
(IIJi
2
($00)
(0 2*
(lI~‘lJ5
f J5)
3
(2J500)
(O$O)
(2/Y/5 I/$
f Jm
4
(200)
(1fiO)
(12/d
5
(2 0 0)
(OflO)
U3lfi
6
(2J5 0 0)
(40
0)
J%
0)
(4lJ3
l/fi
* 218)
f l/J% * VJlo) SI@
* l/J%.
[6, 7, 81 [lo] [ll, 91 [8, 61.
178
KOIWA
e! a/.:
TILT BOUNDARIES IN HARD-SPHERE f.c.c. CRYSTALS Table 3. Displacement vectors
No. 1 2 3 4 5
6
A
A A A A A
7
D A D
a
A
9 10
II 12 13
A
0.3440
A D A
A D A :
16
A
D
19 20
l.Wlf
D
A A D
18
0.4605
0.2727 2.7942 0.7071 0. 0.3903 0.7071
14 IS
17
O.fO?l 0. 1.7321
0.1940 0. 0.9303 0.6417 1. 1.4334 I.1547 I.4891 0. 0.1340 0.7854
0.7071 0.
A
0.
:
0.7071
A
it
:
0.
A
0. 1.53% 1.0300 0.7698
A
D G
0.4082-1.t547 1.7321-0.6719 -0.6719 0. I .0297-0.8530 1.581ko.7187 0.1424-0.7402 I .2533-0.9708 I. -0.7071 1.7961-0.8649 2.1213-0.6720 0.7071-l. 1.1408-0.7617 3.3166-0.6433 t.ao9t-o.a528 1.~~.78~ 2.5495-0.7223 0.1%1-0.9806 0.2175-0.7536 0.2265-0.784s 1.9378-0.9359 1.2127-0.7276 2.9155-0.7114 1.251~.82~ 2.PlfS-0.6264 1.0290-0.9701 4.1231-0.6101 2.9913-0.7432 0.9701-1.0290 2.4374-0.7297 2.~7~.7~8 4.3589-0.8509 1.2329-0.7620 1.3424-0.7030 1.779!&0.7315 0.8117-0.8652
B ::
0. I.1547
B B
kl277 0.9017
B
0.7071 0.7698
0.769a-O.9526 0.7071-0.9526
E
I. 1.1547
0.3 162-O.po87
J.0801-0.7776
0.707 1
1.3224-0.7533
c
0.
3.
B
0.
1.7641-0.7423
c
0.
1.0999-0.8889
B B E ::
0.7071 0.7071 0. 00:7226
0.9239-WMO 2.7526-0.7500 l.ooso-o.9949 0,887~.a3# 1.1767-o.7845
C F
0.7071 0.7071
:
;:77o4
2.2111-0.7817 0.6030-1.0660 0.5547-o.8322 1.9642-0.7937
c
0.6773 1.7240
0.9624-0.7843 1.47209.8974
0.4472-0.8944
0.8474 1.3ooa
0.8339-0.7658 0.8286-0.7937
8”
0: 05609
E
0.6294
! E”
0.7071 0.7071 0.
H
0.7071
;
II:7071
1.8106-0.8067 2.669!W.?263 0.9~.~78 2.2293-0.7276 0.6288-1.0510 3.62lO-O.7101 2.182a-o.8575 0.5659-1.0862 2.3099-0.7380 0.9177-o.8111
B B E H
0. 1.5259 1.0755 2.5852
1.1896-0.7647 1.3219-O.7043 1.3313-O.7532 ~1.3M8-O.9421
5. I. Comparison with boundury structures by computer ~~uiut~n
It is interesting to compare the atom configurations of grain boundaries obtained by computer simulation and those predicted by the present analysis with IAM. For the [1001(013) boundary, Hashimoto ef al. [l’2] have found two types of structures, Type I and Type 2, the former one is similar to those reported by Weins et al. [6] and Smith el a/. [8]. The structures shown in Fig. 6(a) and (b) are very similar to Type 1 and that in Fig. 6(c) to Type 2 (compare with Figs 3 and 5 in Ref. [12]). The compa~~n can be made more clearly by plotting the displacement vector corresponding to each simulated structure in the map, as shown in Fig. 8. The vectors are located very close to those derived in the present analysis.
scheme.
is translated
1.1772-0.7732
B E
5. DISCUSSION
present
c C
In these figures the projection of the vector d, is shown by the broken line. Note that the vector I$ is not generally orthogonal to the vector dt; the area of the rectangles in the figures is two or three times of that of the repeat unit spanned by d, and d2.
Vhe x, v and z terminology
1.1547-0.8165 -0.8165 0.
to conform
to the
F
-0.7917
;:
O.2425-0.9701 2.1671-0.8266 0.5145-0*8575 1.8293-O.8085
0.0900 0.
3.3485-0.7246 1.7786-o.9147
0.1834 0.7071
2.7284-0.7532 1.7816-o.8314
0.7071
1.6244 1.1547
0.8113-0.9177 0.0829-0.7la1 0.0937-0.811 I
2.8587
0.8246-0.9765
;:
: F I
1.t5435.80~8
5.2. The determination of initial con&rations
The procedure for determination of initial previous investigations (1) Weins et al. [6],
the rigid translation or the configurations adopted by the are: (WGC)?
l‘. . . the crystals were translated from their coincidence configurations in the ydirection and in the x-direction in increments of l/20 of the period in the y-direction, and l/10 of the period in the x-direction. At each of the y- and x-translations the energy minimum by relaxation in the z-direction was determined using an increment in z of 0.1 A which corresponds to a step of approximately l/5 of the minimum step size in the y-direction.. . .‘I’ (2) Smith et al. [8], (SVP) l‘. . . The procedure adopted was to begin with the coincidence structure and allow this to relax.. . . Next, the first plane of atoms parallel to the boundary was removed from the upper crystal.. . . In this way, up to 6 planes were removed su&essivdy from the upper and the lower grains before relaxation was carried out.. . .” (3) Hashimoto et al. [12], (HIYD) 6, . . , The upper grain was moved relative to the lower grain by small increments parallel to the boundary plane without rotation and maintaining the atoms on the original lattice sites of their respective
KUIWA
et aI_: TILT BOUNDARfES
d, =‘(200),
IN HARD-SPHERE
d,=(k!),
d, =(-ioi
kc.
CRYSTALS
1
(4
d, =(2221,
d,=f301),
d,-f0ii)
d, =(IT0t
d, =( 110) , (c)
d, -(200),
cl,‘tt231,
d3=fOlTi
Fig. 7. The distribution maps of the displacement vectors for several boundaries. Note that the recta&s arc not the repeat unit of respective boundaries in these examples. The displacement vectors giving rn~~n~~lly stable ~n~gu~tions were sought over a half or a one-fourth of the repeat unit spanned by the vectors d, and d,. The vector d, is appropriately displaced so as to point to inside the rectangle. (a) Sequence [4] [1OO](OZl)ZS,(b) Sequence 6 [111](123)1;7 (c) Sequence 9 [I lO](l13)211. (d) Sequence 11 [lOO](O32)r;13.
179
180
KOlWA
et al.:
TILT ~UNDARIES
IN HARD-SPHERE
f.c.c. CRYSTALS
plane. Figure 9 indicates the displacements corresponding to the removal of 0, I, 2,. . . ,Q layers, for the 25 boundary. For each boundary, there exists a number of layers, NT, the removal of which results in apparently different configurations. For the present CS boundary, the number is 10 (see Table 1). AS seen in Fig. 9, the successive removal of layers scans the whole area first sparsely and then minutely; the procedure seems to be very reasonable. Nevertheless, one examines equivalent configurations if the method is mechanically applied. in the case of Fig. 9, the following pairs are equivalent: (0 8), (1 7), (2 6), and (3 5). Note that the marked points in the figure regularly Fig. 8. Comparison of displacnnent vectors determined by align on certain lines. Since the minimum points may previous investigators for the [100](013)25 boundary. The locate at places distant from those lines as seen in Fig. locations marked as W and S are those determined by Weins 8, there might be a possibility of missing the lowest er al. [6] and Smith et ul. [S], respectively. H, and H, refer energy configuration by starting only from those to “type I” and “type 2” in Ref. 1121.The marks,0, A and 0, representthe displacement vectors determined in the configurations produced by the removal of layers. presentstudy. 5.3, Excess volume of grain boundaries crystals. The increments spectively, l/10 and l/30
of translations
were, re-
of the periods in the X- and y-directions.. . . The same process was continued for five steps. In each succeeding step the increments of translation were made smaller than that of the previous step. . . . in order to determine the minimum energy configuration, the above calculation process was started from various structures made by removing or adding a ‘few atomic layers from or to the Z5 coincidence lattice site ~lations~p . . .”
The difference between the volume of a bicrystal and that of a perfect single crystal is the definition of the “excess volume” associated with the grain boundary. Frost et al. [4,5] have caIculated the excess volume, Vsx of the densest configurations for 51 symmetric coincidence tilt boundaries. We have extended the calculations and obtained the values for total 98 boundaries (CfC103). The values are plotted as a function of the area of the repeat unit, as shown in Fig. 10. The excess volume increases with the increase in the area of the repeat unit, and seems to have a station value less than 0.5, which co&rrns the trend observed by Frost et. al. [5j. It is added here that we have compared our results with those given in the “catalogue” by Frost et al. 1131,for all the non-faceted densest boundaries with C-values less than 45; an excellent general agreement is found as it should be. One exception is the
In the light of the result of the present investigation, some remarks are made on the efficiency of the above described procedures for the determination of starting configurations, The procedure of drawing a grid over a certain area and calculating the energy for each grid point, adog ted by WGC!and HIYD, seems to be a natural one; the finer the mesh size, the better one has the opportunity of encountering a minimum energy point. In adopting this way of the search, it is worthwhile to remember that only a half (the [ill] rotation axis) or one fourth (the [lOOI and [lIOJ rotation axes) of the area of the repeat unit is to be examined. At each of the y- and n-translations, WGC tried to find the energy minimum by allowing the relaxation in the z-direction, while HIYD seem to have calculated simply the energy for a fixed z-translation. The present hard sphere analysis is similar, in some sense, to the WGC analysis, and is expected to yield results with a good correspondence to those by WGC but not to those by HIYD. SVP produced different initial configurations by Fig. 9. Displacement vectors ~uiva~ent to the removal of successiveremoval of layers followed by the shift of layers for the constriction of initial structures for computer the relevant crystal by one interplaner spacing, which is equivalent to certain displacements in the x-y
simulation. The numerals in the figure correspond to the number of layers removed.
KOIWA er al.:
TILT BOUNDARIES
IN HARD-SPHERE
f.c.c. CRYSTALS
181
0.60
0 30 t 0
0.20 0
0.10 ~
0.01 + 0
2
’
’
’
4
6
B
’
f
’
’
’
’
’
’
’
t
1
10 12 14 16 I8 20 22 24 26 28 30
Area
of repeat
unit
Fig. 10. Excess volume as a function of the area of the repeat unit of boundaries.
[l 10](115)~27; our v, is 0.3055 while theirs is 0.3053. As to the sizes of substitutional or interstitial holes given in their catalogue, minor differences with our results were noticed only for the [100](032), [I 11](145), and [100](043) boundaries. 6, CONCLUSIONS AND SUMMARY
A systematic method is developed for the study of the symmetric tilt boundary models between hard sphere f.c.c. crystals. The analysis begins by finding a proper set of basis vectors in describing the atom arrangement in crystals containing a boundary. The problem of finding mechanically stable and/or dense configurations of two crystal blocks can be simplified by introducing the concept of “image” atoms. The atom arrangement in one block is overlapped with that in the other, so that the problem is reduced to the positioning of a single atom sphere relative to one crystal block containing both real and image atom spheres. The new method, referred to as the image atom method (IAM), allows a systematic, yet economical and thorough search for the mechanically stable configurations. The method has been applied to 98 symmetric coincidence tilt boundaries with [loo], [llO] and [ll l] tilt axes, and numerical results are given in tabulated forms for some boundaries. There are generally several stabIe configurations for a given boundary. Some boundaries have as many as 20 different stable configurations or the displacement vectors. The derived structures were compared with those obtained by computer simulation with a qualitatively good agreement. In the light of the present analysis, some remarks have been made on the validity of the procedure, adopted by previous investigators, for the choice of initial configurations for computer simulation. It is concluded that a careful analysis of the geometry of
the hard sphere model of boundaries provides useful information for more advanced investigations by using realistic interatomic potentials.
~~~~e~~e~~f~-~e authors are greatly indebtecl to Dr S. Ishioka for useful advice and eniightening discussion. REFERENCES
Harrison, G. A. Bruggeman and G. H. Bishop, &a& Boundary Structure and Properties (edited by G. A. Chadwick and D. A. Smith), p. 45. Academic Press, New York (1976). 2. V. Vitck, A. P. Sutton, D. A. Smith and R. C. Pond, I. R. J.
GrainBounaiuyStructureand Kinetics,p. I 15. Am. Sot. Metals, Metals Park, OH (1980). 3. V. Vitek, R. C. Pond and D. A. Smith, Computer Simulationfor Material Application, N&ear Metalkg-y, Vol. $0, p. 265. Plenum Press, New York (1976). 4. H. I. Frost, M. F. Ashbv and F. Snaenen. . . . Scrivta . metall. 14, iOSl (1980). * 5. H. J. Frost, F. Spaepen and M. F. Ashby, Scripro metall. 16, 1165 (1982). 6. M. J. Weins, H. Gleiter and B. Chalmers, f. u&. Phys. 4% 2639 (I 971). 7. 0. Hasson, J.-Y. Boos, I. Herbeuval, ,M. Biscondi and C. Goux, Surf. Sci. 31, I15 (1972). 8. D. A. Smith, V. Vitek and R. C. Pond, Acta metall.25, 475 (1977). 9. N. Hashimoto, Y. Ishida, R. Yamamoto and M. Doy-
ama, Acta met&f. 29, 617 (1981). IO. G. Hasson, M. Biscondi, P. Lag&e, J. Levy and C. Goux, The Nature and Bet&or of Grain 3ounalaries (editui by H. Hu), p. 3. Plenum Press, New York (1972). 1 I. R. C. Pond, D. A. Smith and V. Vitek, Acta metoll.27, 235 (1979). 12. M. Hashimoto, Y. Ishida, R. Yamamoto and M. Doyama, J. Phys. F, Metal Phys. 10, 1109 (1980). 13. H. J. Frost, M. F. Ashby and F. Spaepen A ~ruIogue of [ ftX& f [lo], and [tl I] Symmetric Tilt Boundariesin Face-Centered Cubic Hard Sphere Crystals. Harvard Univ. Press (1982). APPENDIX In the s~rne~c
tilt grain boundaries of the hard-sphere model, spheres on the first layer are always in touch with spheres belonging to the counterpart block, thus determining the boundary separation. This can be proved by showing that the following situation cannot happen. “A sphere P in the mth layer of the upper block is in contact with a sphere Q in the nth layer of the lower block, where m z n > I.” For simplicity’s sake, we adopt a new coordinate system explained in Section 2.3, and further take the x-axis along
KOIWA er al.:
182
TILT BOUNDARIES
the d, direction. Then the vectors d;, d; and the displacement vector T’ for the relevant boundary are written as
IN HARD-SPHERE
Similarly, in the lower block a sphere Q’ exists at -(n - 1)d; from Q, i.e. in the first layer Q’ = Q - (n - 1)d;
d; = (x0, yo. ~a), d; = (xgl YO.-20) T’ = (XT, YT. -IT),
IT >
= (x2 - (n - t)-%, Y2 - (n - IlYo. -+I.
0.
The square of the distance p’e’
The coordinates of spheres P and Q are YI, z,)=(x,,
P=(x,,
Q = (x,, YZ, 23 = h
f.c.c. CRYSTALS
is
(p’ - Q)2 = (x, - x2)2
YI. (m - lb,)
+ (_Y,-yd2 + [(m - n)za + zJ2.
~2, -61 - llz,, - zT).
(A2)
By comparing with equation (Al)
Since the two spheres are in contact, one has
(I” - Q’)2 < 4r,. (P-Q)2=4r: or
(Al)
(x, --x*)2 + (_I$- YJ’ + [(m + n - 2)za + ZJ = 4G I where r, is the sphere radius. In the upper block, there exists a sphere P’ at -(a - l)d; from P, i.e. in the (m - n + 1)th layer, P=P-(n-l)d; = (x, -(n
- 1)x,, YI -(n
- l)Yo, (m - nk).
Evidently, this cannot happen, since spheres cannot overlap each other in the hard-sphere model. In order to satisfy the relation, (P’-Q)224rf or [(m - n)za + zrl’ 2 [(m + n - 2)~s + 2$, n must be unity. This proves the statement at the beginning of Appendix.