HAL Id: jpa-00216314
https://hal.archives-ouvertes.fr/jpa-00216314
Submitted on 1 Jan 1975
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
EXTENSIONS FROM MODELS OF SIMPLE SHORT PERIOD GRAIN BOUNDARIES
M. Weins, J. Weins
To cite this version:
M. Weins, J. Weins. EXTENSIONS FROM MODELS OF SIMPLE SHORT PERIOD GRAIN BOUNDARIES. Journal de Physique Colloques, 1975, 36 (C4), pp.C4-81-C4-86.
�10.1051/jphyscol:1975409�. �jpa-00216314�
JOURNAL DE PHYSIQUE Colloque C4, supplkment au no 10, Tome 36, Octobre 1975, page C4-81
EXTENSIONS FROM MODELS OF SIMPLE SHORT PERIOD GRAIN - BOUNDA-RIES
M. J. WEINS
Department of Materials Engineering University of Illinois
Chicago, Illinois. USA and
J. J. WEINS Material Science Division Argonne National Lziboratories
Argonne, Illinois, USA
RCsume. - L a structure e t 1'Cnergie d e deux joints intergranulaires, avec des angles d e 36,9", entre les plans (100) pour l'un e t entre les plans (1 10) pour I'autre, ont CtC calculCes pour I'or et trois alliages or-cuivre.
L e s trois alliages Ctaient : a) un alliage ordonnb 50-50,
b) u n alliage d'or contenant un atome d e cuivre pour chaque pCriode d e la structure d e joint, c) un alliage biphasC compos6 d'un grain d e cuivre pur e t d'un grain d'or pur s6parCs par un joint d e grains.
Si, dans l e systkme Cu-Au, caractCris6 par une solubilitC complkte, les structures d e s joints d e grains trouvCes ne sont pas sensibles B la composition d e l'alliage, les atomes Ctrangers ont cependant tendance a sCgrCger a u joint, dans des alliages diluCs.
Plusieurs modbles d e structure des joints intergranulaires sont discutCs et une nouvelle description d e ces structures est proposCe.
Abstract. - T h e structure and energy of two 36.9" grain boundaries, one between (100) planes xnd the other between ( 1 10) planes, were c;~lculated for gold and three gold-copper alloys. Three types of alloys were examined : (a) an ordered alloy containing a n equal number of gold and copper atoms, (b) a gold rich alloy i n which only one copper impurity atom was associated With each periodic unit of boundary structure, and (c) a t w o phase alloy with an interphase boundary between a pure gold grain and a pure copper grain. In this alloy system, which is characterized by a high degree of solubility, the grain boundary structures were not found t o be sensitive to the alloying conditions ; however, there was a tendency in dilute solution for foreign atoms t o segregate t o the grain boundary. Several of t h e existing methods of describing the grain boundary structure are discussed and a new :rltern;~tive method of describing t h e structure is proposed.
1 . Introduction. - All of the previous computer modeling work on atomistic grain boundary structu- res has been done on grain boundaries in pure materials. This work was undertaken to determine the grain boundary structures in three types of gold-copper alloys : (a) an alloy in which the grain boundary is between ordered crystals each of which contain a n equal number of gold and copper atoms ; (b) an alloy of gold in which the atomic fraction of copper is sufficiently small that. only one copper impurity atom is associated with each unit of grain boundary structure ; and finally (c), an alloy in which the grain boundary is an interphase boundary between a copper and a gold crystal. The minimum energy grain boundary structures for the alloys were compared to the structure of the correspond- ing pure gold grain boundaries. The gold-copper
system was selected because there is a high degree of substitutional solubility and an ordered phase exists.
Two different boundary geometries were selected for the study : (i) a boundary with the (100) planes symmetrically disposed from the plane of the boundary, and (ii) a boundary with the (1 10) planes symmetrically disposed from the boundary plane.
The relative rotation of the crystals about a [001]
axis for both geometries was by 36.9".
2 . Procedure. - The procedure used t o generate these structures was the same procedure a s was used by Weins [I] in determining the structure of boundaries in pure materials. The first step was a block relaxation to determine the relative orienta- tion of the two crystals associated with minimum
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1975409
C4-82 M. J . WEINS AND J. J. WEINS
energy. In this step the two crystals were moved relativebto each other in the boundary plane, both parallel and perpendicular, to the tilt axis in increments that were small with respect t o the period of the boundary. The increments were chosen t o be a small fraction of the boundary period. These increments were not fractions of lattice vectors of any special sublattice as suggested by Bruggeman and Bishop [2]. Work done by Bamiro, Weins and Weins [3] has indicated that further subdivision of the steps only slightly modified the energy and in general did not result in translations that were vector translations of any sublattice. In the second step the atoms in the bicrystals, that were formed using the translations that were associated with minimum energy, were allowed t o relax from their lattice sites to sites associated with minimum energy. The resulting structures were then checked to see if voids greater than 0.9 of an atomic diameter existed. If such voids were found, additional atoms were added t o the structure and the energy recalculated to deter- mine if a lower energy structure could be obtained.
The atomic interactions were represented by a six-twelve potential. The method of determining the potential constants was that of Lennard-Jones for the pure materials [ 4 ] . For the Au-Cu interactions a pseudo heat of sublimation based on the root mean square of the gold and copper heats of sublima- tions, and an arithmetic mean separation were used.
This method is described in detail by Tick [5]. The form of the potential and the constants for the three types of interactions are given in table I. The six-twelve potential was selected for this work on the basis that the structure and energy terms are separable. This facilitates the ability to construct reasonable and comparable potentials for the mate- rials.
The energy of the bicrystal was obtained by summing the interactions of each atom with all neighboring atoms that were located a t a distance less than the separation of the second nearest neighbor in a perfect crystal. The sum was divided by 2 to correct for the double counting of atomic interactions.
3 . Resutts. - The relative translations associated with the minimum energy structures for the four alloys in which the (110) planes are symmetrically disposed from the grain boundary plane are given in table 11. All of the minimum energy structures are associated with translations from the original boun- dary coincidence configurations. The minimum energy boundary in pure gold and in the gold alloy containing a small atom fraction of copper have the same relative translations. The boundary in the ordered alloy has a z component to the minimum energy translation which was not present in the gold, dilute gold alloy, and the interphase bounda-
Form of the 6-12 potential used in this work
where *(rij) is the potential energy between atoms i and j at a separation of rij and
whence the subscripts q, p denote the type of atoms interacting ; 1 for an A-atom ; 2 for a B-atom. In the case where there are two different atoms designated as A and B atoms :
L = heat of sublimation L'2 = UL-2
aI2 = 1/2(a1'
+
a22)The constants are as follows for the case involving gold and copper atoms :
C,,, = a crystal potential constant (geometric factor)
For an fcc material C6 = 14.4589, CI:! = 12.1313
Gold : Sublimation energy, L22 = 87.3 kcallg-atom E q u i l i b r i u m a t o m i c s e p a r a t i o n , a22 = 2.878
A
Copper : Sublimation energy, L1' = 8 1 . 1 kcallg- atom
E q u i l i b r i u m a t o m i c s e p a r a t i o n , a" = 2.857
A
ries. The ordered alloy and the interphase bounda- ries have a different x translation than the pure material, however, the boundary has a structure similar to that of a pure material. This type of structure equivalency is discussed elsewhere in detail by the author [ I ] and results from symmetry which is preserved in translation. Table I11 gives the relative translations associated with the mini- mum energy structures in the boundaries that have the (100) planes symmetrically disposed from the boundary plane by 36:9O. For these boundaries the translations are identical for the pure gold boundary and the boundary containing the copper impurity, and are only slightly different for the ordered alloy and the interphase boundary. This is in part due to the fact that the lattice parameter in the alloys
EXTENSIONS FROM MODELS O F SIMPLE SHORT PERIOD GRAIN BOUNDARIES C4-83
(1 lo), 36.9" boundary translations resulting in minimum energy configuration
Boundary X-Translation Y-Translation Z-Translation p e r i o d - ~ / ~ e r i o d - z
Pure Au 5.9
Au with Cu impurity 5.9
Ordered Au and Cu 1.4
Interphase boundary 8.6
(loo), 36.9" boundary translations resulting in minimum energy configuration
X-Translation Y-Translation Z-Translation Period-XIPeriod-Z
- - - -
Pure Au 5.1 0.8 2.0 6.43514.07
Au with Cu impurity 5.1 0.8 2.0 6.43514.07
Ordered Au and Cu 5.1 0.8 2.0 6.41214.055
Interphase boundary 5.1 0.8 2.0 6.43514.07
differs from the lattice parameter in the pure material.
Figures 1 through 5 depict representative grain boundary structures for the alloys studied. Figure 1 shows the minimum energy structure for a pure gold boundary in which the (110) planes are symmetrically disposed. It is apparent from an examination of this figure that there is a high degree of path connectivity between atomic planes in one crystal and alternate planes in the adjacent crystal.
FIG the
. 1. - The structure of a pure gold grain boundary in which (110) planes are separated by 36.9" about a [001] axis. The
dotted lines indicate the paths of connectivity.
This connectivity is illustrated by the dotted lines.
The figure shows that there are some shared atoms ; however, the atoms are not equally shared by the two crystals as was proposed by Bishop and Chalmers [6]. Figure 2 shows the minimum energy structure for a pure gold boundary which has the (100) planes symmetrically disposed. The connecti- vity of planes in the crystals is again evident and a similar interpretation may be employed. In this boundary, the planes in one crystal are connected with every third plane in the second crystal. This is illustrated by the dashed line.
FIG. 2. - The structure of a pure gold grain boundary in which the (100) planes are separated by 36.9" about a [001] axis. The dashed lines indicate the paths of connectivity of every plane in
one crystal with every third plane in the other crystal.
Figure 3 shows a boundary symmetrically dis- posed between (1 10) planes in a gold bicrystal that contains a single copper impurity atom in each unit
F I G . 3. -The structure of a grain boundary between a pure gold crystal and a gold crystal containing a copper impurity atom in which the (1 10) planes are separated by 36.9" about a [001] axis.
The copper impurity atom is indicated by the solid circle and is in the energetically most favored position. The interpenetrating
ledges are indicated by the lines.
C4-84 M. J. WEINS AND J . J. WEINS
of boundary structure. The copper impurity atom is indicated by a solid circle. This boundary structure may be interpreted a s was done for the preceding boundaries or an alternative description of interpe- netrating ledges could be employed as has been suggested by Weins, Gleiter and Chalmers [7]. The interpenetrating ledges are indicated by the lines.
Figure 4 shows the minimum energy structure of an interphase boundary between a gold and a copper crystal. The gold crystal is indicated by the open symbols, and the copper by the solid symbols.
A comparison of this figure with the previous figure will show that the structure of this interphase boundary is quite similar to the structure of a boundary in pure gold ; and this boundary could be described in the manners discussed with figures 1 and 3. In addition, the dislocation structure descrip- tion proposed by Read and Shockley [8] could be used. This description is illustrated by the symbols that are present in the boundary indicating disloca- tions.
FIG. 4. - The structure of an interphase boundary between a gold and a copper crystal in which the (1 1 0 ) planes are separated by 36.9O about a LO011 axis. The gold crystal is indicated by the open symbols, while the copper crystal is indicated by the solid symbols. The dislocation network that could be used to describe
the grain boundary structure is indicated.
Figure 5 illustrates a minimum energy grain boundary structure in an ordered alloy containing equal portions of both gold and copper. Again, the structure is very similar to the structure associated with the pure material, and the structure of this
FIG. 5. - The structure of a grain boundary in an ordered alloy containing an equal number of gold and copper atoms, and in which the (1 1 0 ) planes are separated by 36.9O about a [OOI] axis.
The gold atoms are indicated by the open symbols, while the copper atoms are indicated by the solid symbols.
FIG. 6 . - The structure of a grain boundary between two gold crystals in which the ( 1 1 0 ) planes are separated by 36.9O about a [OOI] axes. Copper impurity atoms were introduced in the positions numbered and the energy was calculated for each location. Since two periods are shown the prime positions are
the equivalent positions in the second period.
boundary could be described using any of the three methods discussed above. In order to determine if there was a tendency for a foreign atom to segregate, the energy of a section of grain boundary was determined with a copper impurity atom in different positions. Figure 6 shows the different locations in the grain boundary segment that were occupied by the copper impurity atom and table IV lists the energy change associated with each position. From an examination of table IV it can be seen that position 6, situated at the boundary, has the lowest configurational energy (most negative) and is therefore a more highly probable site and can be described a s a position of least strain. Also, movement of the impurity atom from positions 3 to 1 requires positive energy ; thus translations from position 1 located in the interior to position 3, located at the boundary will be more favorable than counter movement. However, along the boundary positions 4 and 5 are relatively high energy posi-
Energy o f the atomic configurations for the various positions of the copper atom in the 36.9" gold
boundary
Change in energy associated Position of with introducing the copper
Cu-Atom impurity in the indicated position (k callg-atom) -
1 2.25
2 3.70
3 2.22
4 3.72
5 3.72
6 1.91
7 2.46
8 2.47
9 2.21
At great distance
from boundary 4.1
EXTENSIONS FROM MODELS O F SIMPLE SHORT PERIOD GRAIN BOUNDARIES C4-85
tions for impurities ; therefore, while there tends to be segregation of impurities to the boundary, there are certain more favorable positions within the boundary.
4. Discussion and conclusions. - The minimum energy grain boundary structures were determined for four different alloying conditions, as well a s for several different locations of a single substitutional impurity in a gold bicrystal. Three methods of interpreting grain boundary structure were pro- posed, discussed and illustrated ; the method of describing the structure of a minimum energy grain boundary proposed and illustrated in conjunction with figures 1 and 2 was a new method not previously discussed in the literature. This new method considers grain boundaries to be regions in a crystal in which there is continuity of only some of the planes. The angles and type of planes that are disposed from the boundary plane will deter- mine the number of planes which have connectivity and the angle of bend that must be made in connecting the planes. This description is helpful in accounting for the anisotropic diffusion observed in grain boundaries 191, and this description will predict that grain boundary structure is not highly sensitive t o alloying conditions which was demons- trated by this work. The method which describes the grain boundary a s a series of interpenetrating
ledges was illustrated in figure 3 and was discussed earlier by Weins, Gleiter and Chalmers [7]. The method illustrated in figure 4 considers the grain boundary t o be a high density dislocation network.
This description was proposed earlier by Read and Shockley [S]. All of these methods are similar in that they predict a dense packing of the grain boundary region and indicate that the grain boun- dary is an ordered region.
The energy of a gold bicrystal boundary was determined with a single copper impurity in diffe- rent locations. It was found that the most favorable position for the impurity was in the boundary and not in the matrix. This was attributed t o the fact that the bonding at the grain boundary region is less constrained.
This study determined the structure of grain boundaries under four different alloying conditions for a system in which there was a high degree of substitutional solubility. All of the grain boundary structures were similar.
Acknowledgement. - The authors would like to thank 0 . A. Bamiro who helped with much of the calculations, the Computer Center a t the University of Illinois at Chicago Circle for the use of the computer during the early work, and the National Science Foundation for their research initiation grant that supported the work.
References
[ I ] WEINS, M. J., Interatomic Potentials and Simulation of [5] TICK, P., Thesis, M. I. T. (1967).
Lattice Defects (Plenum Press), 1972, 695. [6] BISHOP, G . and CHALMERS, B., Scr. Metall. 2 (1968), 133.
[2] BRUGGEMAN, G. A. and BISHOP, G. H., J. Appl. Phys. 44 [7] WEINS, M. J., GLEITER, H., and CHALMERS, B., J. Appl.
(1973), 4468. Phys. 42 (1971), 2639.
[3] BAMIRO, 0 . A., WEINS, M. J. and WEINS, J. J., to be [8] READ, W. T., and SHOCKLEY, W., Phys. Rev. 78 (1950), 275.
published. [9] HOFFMAN, R. E., Acta Met. 4 (1956), 97.
[4] JONES, J. E. and INGHAM, A. E., Proc. R. Soc. (London), A 107 (1925), 636.
DISCUSSION
G. MARTIN : In the concentrated alloy which you studied did you look for the effect of changing the grain boundary chemical composition ?
M. J. WEINS : An extensive study of the variation of the chemistry of the boundary of the ordered phase was not undertaken since prelimi- nary studies with variations of chemistry did not substantially change the structure.
M. BISCONDI : Comment, de f a ~ o n aussi prCcise que possible, a kt6 dCterminC le potentiel d'interac- tion entre deux atomes de nature diffkrente ?
M. J. WEINS : I have used 6-12 potential which allowed one to seperate the structure factor from the energy factor. I could then weight the structure component by employing an arithmetic average of the equilibrium radius of copper and the equilibrium radius of gold and I could employ a weighted heat
C4-86 M. J . WEINS AND J . J . WEINS
of sublimation of gold and copper. This procedure has been employed by Tick to simulate the interaction between copper and gold. In addition the 6-12 potential gave results in grain boundary structure studies that were similar to the results I obtained when I used a Morse potential.
P. PUMPHREY : You have been talking only about short period boundaries. Would you expect that a long period boundary would contain many atoms of very similar energies, making the ordered segre- tated structures you have shown less likely ?
M. J. WEINS : The calculations to this point have been performed only for short period boundaries.
However on might still anticipate segregation in longer period boundaries since it is believed that the longer boundaries are composed of the same or similar structural units with the missfit accomo- dated by dislocations. It should be noted that the dislocations will offer possible additional segrega- tion sites might make the distribution of impurity more uniform along the boundary.
K . LUCKE : I like to reformulate the question of Dr. Haessner. Can you give the absolute values of the energy changes for a Cu-atom put into the boundary and for a Cu-atom put into the perfect
lattice ? The difference between both should be the interaction energy between the Cu-atom and the grain boundary in gold which would allow t o calculate the enrichment of the Cu-atoms in the boundary.
M. J. WEINS : The values given in the paper are, for a Cu-atom in the boundary : 1.9 keV/Mol and for one far away from the boundary 4.1 keV/Mol.
K. LUCKE : This means the difference is about 2.2 keV/Mol approximatly 0.1 eV/atom, i. e. that is the difference generally assumed.
F. HAESSNER : Can you make some comments on the difference between the concentration of foreign atoms in the boundary region and in the lattice ? How far does the concentration profile extend into the lattice region in your model ?
M. J. WEINS and J. J . WEINS : The concentration in the lattice was essentially zero, while the concentration in the boundary region was suffi- ciently dilute that no interactions between foreign atoms needed to be taken into consideration. Our computer model predicts these enhanced concentra- tions in the boundary region, we beleive this is because of the excess volume and the irregularity of the packing near the boundary.
