Atomic Group Rotation Mechanism for f10�12g Twinningof HCP Crystal Materials

Shan Jiang+

Chongqing Academy of Science and Technology, 2nd Yangliu Road, Huangshan Avenue,New North Zone, Chongqing 401123, China

In this paper the atomic motion law of the f10�12g twinning in hexagonal close-packed (HCP) crystal materials was studied through theatomic group rotation (AGR) model. The research results show that the AGR mechanism has universal property to all the HCP crystal materials.Though the structure of the atomic group changes with the axial ratio, the atomic motion of f10�12g twinning in all the HCP crystal materials canbe ascribed to the rotational motion of the atomic groups. The relation between the rotational angle, the relative displacement magnitude and theaxial ratio was obtained. With the growth of axial ratio, the rotational angle increases, but the relative displacement magnitude decreases.[doi:10.2320/matertrans.M2013472]

(Received December 27, 2013; Accepted March 4, 2014; Published May 25, 2014)

Keywords: hexagonal close-packed, twinning mechanism, atomic group, modeling

1. Introduction

The theory of twinning deformation merits attentionbecause of its inherent importance as a mode of plasticdeformation in many crystalline materials.1­5) For the HCPstructures, twinning is especially important to contribute thedeformation systems required for a general deformation.Although several twinning modes have been observed inHCP materials, the f10�12g mode is the most common one.6­8)

For all likely values of axial ratios (£) in the HCP structures,the f10�12g mode gives the lowest magnitude of twinningshear. Therefore the shear value was believed an importantfactor to determine the twinning modes.9)

Bilby and Crocker proposed the theory of “shear andshuffle”.10,11) They considered the twinning is accomplishedby the cooperation of shearing atoms and shuffling atoms.Recent microscopic study suggested that the twinning isconducted by partial dislocations.12,13) Although researchershave proposed a number of opinions to explain the twinningmechanism, the main idea is that the twinning is accom-plished by the cooperation of different type of atoms.

This condition had not changed until the proposal of theAGR mechanism, in which the atomic groups with certainstructure are regarded as the basic research units.14,15) Fromthe viewpoint of this mechanism, a clear regularity of atomicmotion during twinning is presented. However, the mechan-ism was originally proposed just for magnesium because it isthe most common HCP metal with axial ratio £ (1.623)closest to the ideal value of the rigid sphere close packing(1.633). In this paper, the mechanism is extended to otherHCP structures to make it widely applied, such as Zn Be andTi. During this process the key problem we meet is thevariation of axial ratios, which influences not only the shearvalues but also the structure of the atomic groups. Therefore,the axial ratio value was calculated as a variable quantityother than a fixed value, and then a set of universal AGRmechanism for the whole HCP structures was obtained. Sincethis study is quite fundamental, it is of great scientific

significance for the further study on twinning deformationmechanism.

2. Calculation

A three-dimensional Cartesian coordinate system isconstructed so as to be convenient for the calculation of thespace vectors (Fig. 1). The movement of twinning atoms isillustrated in Fig. 2. As the atoms of HCP structures arestacked in a sequence of “+ABAB+”, the f10�12g twinningplanes are divided into two categories: main planes (M) thatthrough the A-layer atoms and sub-planes (S) that through theB-layer atoms. In addition, the pair of M-plane and S-planeon the f10�12g twin boundary are numbered as “0” and fromwhich to the distant “1, 2, + n” (where n is the sum of thetwinning planes). Thus the k-th main plane is expressed asMk-plane; the rest can be deduced by analogy. All the atomsin each twinning plane move in the same way and each atomviewed in Fig. 2 represents a row of close-packed atomsalong the x-axis direction. Since the appearance of the“shearing” atoms present a periodicity by each four layers,we just select a part of representative atoms of four adjacentplanes to study. A part of the atoms in Fig. 2 and the planesthey belong to are listed in Table 1.

We select the parallelogram “ABCD” (Fig. 2) to be aspecific research area. The displacement vectors of twinning

Fig. 1 Three-dimensional Cartesian coordinate system and the correspond-ing four-axis coordinate system in one HCP cell.

+Corresponding author, E-mail: 382595277@qq.com

Materials Transactions, Vol. 55, No. 6 (2014) pp. 907 to 910©2014 The Japan Institute of Metals and Materials

atoms can be calculated according to the symmetric relationbetween the matrix and the twin. Atom O keeps at theoriginal position for it is at the twin boundary (in the M0

plane). The displacement vector of atom N can not bedetermined for they are so close to the twin boundary (in theS0 plane), hence atom C (in the S2 plane) is calculated insteadfor their periodic relationship. The displacement vectors ofthe twinning atoms are as follows:~vA ¼ ~vM2 ¼ �xAiþ�yAjþ�zAk

¼ 0iþffiffiffi3

pcos 2ªajþ ð

ffiffiffi3

psin 2ª � £Þak ð1Þ

~vB ¼ ~vS1 ¼ �xBiþ�yBjþ�zBk

¼ 0iþffiffiffi3

p

6þ

ffiffiffi3

pcos 2ª

2

� �aj

þffiffiffi3

psin 2ª

2� £

2

� �ak ð2Þ

~vC ¼ ~vS2 ¼ �xCiþ�yCjþ�zIk

¼ 0iþ 5ffiffiffi3

pcos 2ª

6þ

ffiffiffi3

p

6

� �aj

þ 5ffiffiffi3

p

6sin 2ª � £

� �ak ð3Þ

~vM ¼ ~vM1 ¼ �xMiþ�yMjþ�zMk

¼ 0iþffiffiffi3

pcos 2ª

3ajþ

ffiffiffi3

p

3sin 2ª � £

2

� �ak ð4Þ

�xA ¼ �x0A � xA; ð5Þwhere ~vA is the displacement vector of atom M; ~vM2 is thedisplacement vector of atoms in the plane M2; a, c and £ are

the cell parameters of the HCP structures, and £ = c/a; xAand x0A are the partial coordinate of atom A before and aftertwinning; ¦xA is the vector components of atom A along xaxis; and ~vA is the minimum shearing vector of all theshearing atoms (others can be analogized).

j~vAj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�x2A þ�y2A þ�z2A

p¼ a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3þ £2 � 2

ffiffiffi3

p£ sin 2ª

pð6Þ

Thus the twinning shear s1 is obtained:

s1 ¼j~vAj2d1

ð7Þ

here d1 is the interplanar spacing of the f10�12g planes, whichsatisfies the following relation:

d1 ¼ffiffiffi3

p£a

2ffiffiffiffiffiffiffiffiffiffiffiffiffi3þ £2

p ð8Þ

Then according to the periodicity of the twinning atoms weget the general formulas for calculating the displacementvectors of all the twinning atoms:

~vMð2kþiÞ ¼ ~vMi þ k~vA ð9Þ~vSð2kþiÞ ¼ ~vSi þ k~vA ð10Þ

where I = 1, 2; k = 0, 1, 2 + n (n is the sum of the twinningplanes in the twin).

3. Modelling and Discussion

The atomic motion of the twinning seems disorderly for thedifferent movement directions of shearing atoms and shufflingatoms. In general, it was believed that twinning is a coop-erative motion of the shearing atoms and the shuffling atoms.

In this paper we consider the atomic groups with certainstructure composed of shearing atoms and shuffling atoms asa whole to study. The selection of the atomic group followsthe rule that the atomic relative position of the unit keepstable during twinning. As an example, the atomic group“ABCD” (Fig. 2) can be selected as an atomic group unit.The structure of the atomic group is illustrated in Fig. 3,which is composed of millions of octahedrons with theiredges connected. The core idea of the AGR mechanism isthat the twinning is accomplished by the rotation of theatomic groups. As shown in Fig. 4, all the atomic groupsmake a combined motion of shifting and rotation. On theother hand, all the atomic groups make the same motionrelative to the surrounding atoms.

The rotational angle of the atomic groups satisfies thefollowing equation:

¡ ¼ arctan�zM

jyM � yOj ��yM

¼ arctanjz0M � zMj

jyM � yOj � jy0M � yMj¼ arctan

2 sin 2ª � 3 tan ª

3� 2 cos 2ª

¼ arctan

2 sin

�2 arctan

ffiffiffi3

p£

3

��

ffiffiffi3

p£

3� 2 cos

�2 arctan

ffiffiffi3

p£

3

� ð11Þ

Fig. 2 Illustration of atomic motion for f10�12g twining in HCP structures.The colorful spheres denote the atoms before twinning; the white spheresdenote the atoms after twinning; and the arrows denote the displacementvectors of the atoms (similarly hereafter).

Table 1 Atoms in various f10�12g planes of HCP structures.

planes M0 S0 M1 S1 M2 S2 M3

atoms O N M B A C D

S. Jiang908

where ª is the angles between the f10�12g planes and {0002}planes, which satisfies:

ª ¼ arctan

ffiffiffi3

p£

3ð12Þ

When the atomic groups rotate during the twinning, therelative displacement among the atomic groups are produced,and the difficulty degree of twinning should be decided bythe resistance generated in this process. The relativedisplacement ð~­ Þ between the atoms of unit “ABCD” andthat of its adjacent groups can be calculated as follows:

~­ ¼ ~vE � ~vD ¼ ~vA � ~vM

¼ 0iþ 2ffiffiffi3

pcos 2ª

3a jþ 2

ffiffiffi3

psin 2ª

3� £

2

� �ak ð13Þ

and the magnitude of ~­ is:

j~­ j ¼ affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9£2 � 24

ffiffiffi3

p£ sin 2ª þ 48

p6

ð14Þ

The relations between the rotational angles ¡, the relativedisplacement magnitude j~­ j and axis ratios £ are presented inFig. 5. The rotational angle ¡ increases with the growth ofaxial ratio £, but at the same time the relative displacementmagnitude j~­ j decreases. The opposite tendencies of ¡ and

j~­ j should be due to the change of the atomic group structureresulted in from the variation of axial ratios.

The axial ratio 1.732 is a critical point, on the two sides ofthis value the shearing direction is opposite. When the axialratio is 1.732, the twinning shear is zero. According to theviewpoint of “minimum shear criterion”, the closer twinningshear to this value, the easier twinning occurs. Howeverwhen employing the AGR mechanism to analyze thiscondition, it is found that although the shifting motion ofthe atomic group is zero, the rotation motion is inevitable. Inother words, each atomic group must overcome the resistancefrom the ambient atomic groups to induce the rotation, whichneed the corresponding mechanical stress to impose. Thisdemonstrate that only using the twining shear to measure thedifficulty degree of twinning is insufficient.

Certainly, all the above analysis is just from the angle ofcrystallography without considering other influence factors,such as atomic radius, electronic structure and grain size.Hence only under this condition, the rotational angles ¡ andthe relative displacement j~­ j can be regarded as importantparameters to measure difficulty degree of twinning.

4. Conclusion

The AGR mechanism has universal property to all theHCP crystal materials. Though the structure of the atomicgroup change with the axial ratio, the atomic motion canalso be ascribed to the rotational motion of the atomic groups.The relation between the rotational angle ¡, the relativedisplacement magnitude j~­ j and the axial ratio c/a wasobtained. The rotational angle ¡ increases while the relativedisplacement magnitude j~­ j decreases with the growth ofaxial ratio c/a.

Acknowledgements

This work is supported by National Natural ScienceFoundation of China (No. 51301215), Natural ScienceFoundation Project of Chongqing (No. cstc2012jjA 50016),National Science and Technology Support Program(2012BAF09B04) and Chongqing Post-doctor Support Pro-gram (Rc201334).

Fig. 4 Atomic group rotation (AGR) model for f10�12g twinning in HCPstructure.

Fig. 5 Variation of the rotational angles ¡ and the relative displacementmagnitude j~­ j with the axis ratios c/a.

Fig. 3 Structure of a part of connected atomic groups for f10�12g twinningin HCP structure.

Atomic Group Rotation Mechanism for f10�12g Twinning of HCP Crystal Materials 909

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