COMPUTER SIMULATION OF THE (10(-1)2) TWIN ATOMIC STRUCTURE IN HCP METALS

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COMPUTER SIMULATION OF THE (10(-1)2) TWIN ATOMIC STRUCTURE IN HCP METALS

S. Hagège, M. Mori, Y. Ishida

To cite this version:

S. Hagège, M. Mori, Y. Ishida. COMPUTER SIMULATION OF THE (10(-1)2) TWIN ATOMIC STRUCTURE IN HCP METALS. Journal de Physique Colloques, 1990, 51 (C1), pp.C1-161-C1-166.

�10.1051/jphyscol:1990124�. �jpa-00230282�

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Colloque C l , supplbment au n 0 l , Tome 51, j a n v i e r 1990

S. HAGEGE, M. MORI* and Y. ISHIDA*

:NSCP-Paris and CNRS/CECM, F-94407 V i t r y , France I I S , U n i v e r s i t y o f Tokyo, Roppongi, Tokyo 106, Japan

R6sumd La structure atomique relaxbe d'une macle de deformation (1012) dans les m6taux de structure hexagonale compacte a 6t6 simulbe B partir d'un programme base sur un potentiel de paires du type Lemard-Jones. Deux configurations originales sont en compbtition pour dBcrire la forme la plus stable de cette macle. Elles sont caract6risBes par une symgtrie maximale pour les complexes bicolores associ6s. L'interface est soit plane, soit microfacettee et elle est soit associ6e B un rniroir pur, soit B un rniroir avec glissement. Dans le cas d'un cristal macl6, une Bpaisseur minimum a Bt6 mise en Bvidence; l'influence d'une . contrainte extgrieure peut faire croitre la macle et peut r6duire cette Bpaisseur minimum.

Abstract The relaxed atomic structure of the (1012) mechanical twin has been investigated by computer simulation based on a Lennard-Jones pair potential. The two original lowest energy configurations which describe the twin are associated to dichromatic complexes with maximum symmetry. The interface is either planar or microfaceted and lies respectively on a pure mirror and on a glide mirror. Twinned crystals are stable only if their thickness is larger than a minimum value. External strain thickens the twinned region and can reduce the value of that minimum thickness.

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Introduction

By combining a geometric model for the coincidence orientation relationship of a grain boundary / l / and the resulting symmetry of the two crystals taken as a composite material l21 it has been possible to describe mechanical twin boundaries in the hcp (hexagonal close packed) metals 131. In the case of the (1012) twin, a dichromatic complex of maximum symmetry was found to be a reasonable starting configuration for the atomic structure of the bicrystal. In order to confirm this result, which was only based on geometrical and structural considerations, a relaxation of the atomic structure and a calculation of the energy of the interface were performed using a Lennard-Jones potential.

If the literature is very flourishing with computer simulation on interfaces in cubic symmetries (cf 141 for a recent review), the hexagonal has only very recently been dealt with 15-81. Moreover, when dealing with the (101 2) twin 15/,/8/, the relaxed structure is shown to be very similar to the traditional geometric representation of the twin 191, a mirror boundary passing through atoms in coincidence.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990124

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We would like to demonstrate that the lowest energy configurations of the (1012) atomic structure are different from the ones previously assumed by similar types of calculation. The lowest energy configurations are directly related to the maximum symmetry complexes reported here above and are characterized by a contraction a t the boundary. In fact

,

two complementary configurations (M,C), with an almost equivalent energy describe the (1012) atomic structure. They differ by the nature of the mirror plane a t the interface (pure or glide mirror respectively for M and C) and the planarity of the boundary (flat or microfaceted).

In addition to the energy and the atomic structure of one single twin boundary, the structure and behaviour of the whole twin (a parent twinned crystal borded by two twin boundaries in a matrix) has been investigated. A twin has to be of a minimum thickness to be stable in an unstrained matrix; there is! a minimum of strain to be imposed on a crystal to nucleate a stable twin. For a larger strain the twinned crystal will grow in thickness by a simple movement of atoms; i t can be described by a sequence of dislocation glides parallel to the boundary plane and a minute relaxation of the structure. This mechanism, which satisfies a high symmetry of the bicrystal and a structurally acceptable displacement of atoms, is believed to be far more acceptable one than the traditional "shufle" of atoms usually required.

Furthermore experimental evidences of dislocation behaviour in and at the vicinity of the twin boundary plane confirm the model. The "corrugated nature of the boundary plane makes the generalisation of this mechanism straightforward for all (hOfi1) twins.

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Geometric model of the (1012) twin

Figure 1 is the projection of the (1012) twin on the shear plane (1120) with the boundary plane, a mirror of the bicrystal, passing through atoms in coincidence. The set of these two structures, fully characterized by a fixed orientation relationship (pure rotation) and a rigid body translation, is a dichromatic complex 1101. This particular complex is defined by a colored point group, m12'm, and an order of symmetry 4. For a different rigid body translation, the symmetry (belonging to each crystal) and the antisymmetry (exchanging crystal 1 and 2, noted by ') are modified. Therefore a n infinite number of complexes may define the twin orientation relationship. However they can be classified in a finite number of classes with a maximum symmetry uniquely defined 131. If the true character of a twin is the presence of a mirror a t the interface, any rigid body translation perpendicular to the boundary plane will create a new dichromatic complex with a t least the same order of symmetry. For some particular values of the translation, the order of symmetry of the resulting complex can rise to 8 creating a m'm'm complex. Considering in addition the case of a starting configuration with a glide mirror a t the interface (+ 0.5 [lOll] translation), complexes with m'm'm, m'c'm, c'm'm, c'c'm (order 8) can also be taken into consideration (Fig.2). For these highest symmetry configurations, atoms are no longer in coincidence but the center of symmetry, half way in between the two equivalent metal atoms is now in coincidence. Evidently, in P63mmc describing hcp metals, the center of symmetry has the highest point symmetry and this choice let thetwo atoms of the hcp unit cell play an equivalent role in the orientation relationship.

Figure 1 . Projection of the (10i2) twin on the shear plane (1150).

Figure 2. Partial description of the dichromatic complexes related to the twin orientation. The complex labelled A is identical to the one in figure 1: All the complexes represented on the figure are related by a rigid body translation in the xy plane.

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Comauter simulation for one (1012) twin boundarv.

The program used to simulate the atomic structure and to calculate the energy of a twin boundary is based on a Lennard-Jones pair potential truncated between the 5th and the 6th term. The size of the bicrystal was based on the minimum lattice distance for the z direction [1120], the unit of the coincidence unit cell along the y direction [l0117 and a sufficient number of atomic plane along the X direction (25 pairs of yz planes on each side of the interface).

Therefore, by periodic boundary conditions, it was possible to simulate a bicrystal of reasonnable size. The unrelaxed structure, as defined by our geometric modelisation was directly introduced and each crystal was able to relax freely along X and y but not along z. The case of magnesium (c/a= 1.633) was chosen as a reference.

Figures 3a and 3b represent the only two stable configurations we found for the atomic structure of the (1012) twin boundary. They are labelled M and C and are closely related to the m'm'm and c'm'm complexes; they differ only by an expansion of 1.77 and 1.58 % of the unit along X. As a reference the two complexes mq2'm and m'm'm are related by a contraction of 50

% of the same unit. For M the boundary plane is flat and located on the pure mirror plane of the m'm'm complex; for C the boundary plane has a microfaceted configuration and is located on a b glide mirror half way in between the two c glide mirrors of the c'm'm complex.

The respective energy of the M and C configurations are 0.3724 and 0.3706 ~ / The lower m ~ curve in figure 4 shows the evolution of the energy of the structure after relaxation when a small shift has been imposed along the y direction. The two minima can be clearly seen for the 0.0 and 0.5 translation. In addition, any intermediate translation within the range -0.5 and +0.5 relax onto one of the two stable structures:

for t such as -0.17 e t < +0.17 towards M far t such as -0.5 < t < -0.17 and +0.17 < t +0.5 towards C.

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Figure 3. Relaxed configurations of the (10i2) twin for a ) M type with a flat boundary plane

b) C type with a microfaceted boundary.

Considering their respective energy value, i t is difficult to ascertain if there is a significant difference i n stability between M and C. On the contrary there is a definite difference in domain of stability: a much Iarger transIation has to be imposed on the structure C to make it relax onto the structure M than from M to C. As shown on the upper curve of figure 4, there is a regular and symmetrical evolution of the expansion a t the boundary between the M and C configurations.

Figure 4. Energy (right scale and lower curve) and expansion a1,ong t h e ^X axis (left scale and upper curve) of the relaxed configurations after translation along y. The final position of every open circle shows the tendancy of the system to relax towards M or C.

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Com~uter simulation for a (1012) twin

Using the same program and the same experimental conditions, the structure of a whole twin has been investigated. Figure 5 represents the relaxed structure of what appears to be a (1012) twin of minimum thickness: 3.5 T with ~ = 3 ~ ' ~ a c / ( 3 a ~ + c ~ ) ~ ' ~ . One can perceice the nature of the limiting twin boundaries: the upper one is of the M type whereas the lower one is of the C type.

In order to increase the thickness, the two boundaries migrate outwards and thus changing the nature of each of their habit plane at every pair of planes. For instance the upper boundary plane, M type, migrates upwards (-X direction) to the next pair of plane yz and change from the M to the C configuration, and again to M a t the second pair of yz planes. The very close values in energy of these two configurations can explain the fact that the same boundary plane migrates from one configuration to another.

Figure 5. Relaxed configuration of a minimum thickness twin in an unstrained crystal. Note the different nature of the two boundaries.

After relaxation and if any outside strain is imposed, a twin less than 3.5 T thick disappears and the system goes back to a single crystal. If a n appropriate strain is applied, this minimum thickness is reduced and can reach zero: a 40 % strain imposed on a single crystal nucleates, after relaxation, a twin of 3.5 T thickness. A larger strain nucleates a thicker twin.

If an unrelaxed configuration contains a twin of intermediate thickness (0-3.5 T), an intermediate strain (40-0 %) nucleate the minimum size twin. This is illustrated in figure 6.

These results may help to clarify some recent experimental results on mechanical twins in zinc and other hcp metals (Fig. 7 and ref. 11,121 where pile up's and high density of dislocations were often found a t only one tip of the lenticular twinned crystal. It can be then assured that a high enough concentration of dislocations, a t a certain location in a grain, giving rise to a high enough strain in the crystal structure can nucleate by itself a microtwin from this location, across the matrix. The simple situation which is assumed in a computer model (no impurity, no other defect than the boundary) may be quite far from the reality but it remains reasonable to correlate these experimental observations and the results of the present simulati.on based on high symmetry and .low energy configuration.

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Th

i

ckness ( 31Rac/ ( 3a2 +c2 1'"

U"

i

t

Figure 6. Amount of strain necessary to impose on a microtwin to make it thicken to a livable size.

Figure 7. Pile up's of dislocations a t one tip of a twin in deformed zinc.

As detailed in a previous publication /3/, a microtwin, as shown in figure 5, can be nucleated and can grow by a very realistic movement of atoms induced by the glide of a set of dislocations, parallel to the interface, and followed by a minute relaxation. The glide of one zonal dislocation has to occur in the plane located at the coincident center of symmetry displacing two atoms out of four in the coincidence unit cell.

The remaining two atoms have to be displaced independently but only in a direction parallel to the boundary plane which is rather satisfactory.

Finally, after the glide of a second dislocation, the first two atoms undergo a minute relaxation leaving all together a global displacement along slip planes of the structure (Fig. 8).

Figure 8. Nucleation and growth mechanism.

ACKNOWLEDGMENTS

One of the authors (SH) would like to thank the french CNRS for making possible a sejourn in Japan.

REFERENCES

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3. Haghge S., Acta Meta11.1989,37,1199.

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8. Serra A., Bacon D.J., Phil. Mag. 1986, 54, 793; Serra A., Bacon D.J., Pond R.C., Phil. Mag. 1988, 36, 3183.

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