MOLECULAR DYNAMICS STUDY OF HIGH-TEMPERATURE GRAIN-BOUNDARY STABILITY IN A (100) Σ = 29 BICRYSTAL MODEL

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MOLECULAR DYNAMICS STUDY OF HIGH-TEMPERATURE GRAIN-BOUNDARY STABILITY IN A (100) Σ = 29 BICRYSTAL MODEL

T. Nguyen, S. Yip, D. Wolf

To cite this version:

T. Nguyen, S. Yip, D. Wolf. MOLECULAR DYNAMICS STUDY OF HIGH-TEMPERATURE

GRAIN-BOUNDARY STABILITY IN A (100) Σ = 29 BICRYSTAL MODEL. Journal de Physique

Colloques, 1988, 49 (C5), pp.C5-381-C5-385. �10.1051/jphyscol:1988544�. �jpa-00228041�

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JOURNAL DE PHYSIQUE

Colloque C5, supplkment au nolO, Tome 49, octobre 1988

MOLECULAR DYNAMICS STUDY OF HIGH-TEMPERATURE GRAIN-BOUNDARY STABILITY IN A

(100)

.z

= 29 BICRYSTAL MODEL*

T. N G U Y E N ( ~ ) , S. YIP(^) and D. WOLF

M a t e r i a l s

S c z e n c e D i v i s i o n , A r g o n n e

N a t i o n a l

L a b o r a t o r y , A r g o n n e , IL 60439, U.S.A.

ABSTRACT

A new molecular dynamics program for the investigation of planar

bicrystalline interfaces at high temperature and finite external stress has been developed. The constant-stress (Parrinello-Rahman) method is combined with a Region I-Region I1 treatment of the borders, thus avoiding (i) 3d-periodic border conditions (and therefore the presence of %interfaces in the computational unit cell-) and (ii) unphysical constraints arising from fixed borders. The usefulness of this new "semi-periodic border" treatment is illustrated in an investigation of the high-temperature stability and structure of the 1: = 29 (100) twist grain boundary.

I. INTRODUCTION

In recent studies of grain-boundary structural properties at high tem- peratures1s2 symmetrical tilt bicrystals have been investigated. However, any symmetrical tilt boundary may be considered as a 180' twist boundary3, and such a boundary is quite special because it contains only one atom per lattice plane in the unit cell, a property which makes the system energy particularly sensitive to relative translations of the two grains. From the standpoint of a systematic study of grain-boundary properties, it seems preferrable to consider the more general twist boundaries and concentrate on the role of interplanar spacing, or equivalently the role of the grain-boundary energy3.

Aside from the choice of a tilt vs. twist grain boundary, any computer simulation study of grain-boundary properties is confronted by difficulties associ- ated with the treatment of the borders of the simulation cell. Neither of the conventional treatments, the periodic and the fixed border's5 can be said to be free of criticism. A new border treatment is needed which does not introduce a second boundary as is the case with the periodic border condition, and also does not subject the simulation cell to rigid, unphysical constraints, as in fixed-border treatment. In addition, previous simulations have been carried out under the constraint of constant volume and, although this volume is adjusted to give similar pressure at different temperatures, it would be more desirable to perform the simulation under conditions of constant external stress6.

In this woik we describe the formulation of a Region I-Region I1 border treatment which is an improvement over the conventional treatments, and we report preliminary results at constant stress on interfacial disordering in a (100) X = 29 twist-boundary model which incorporates the new border treatment. The system we stu y is composed of atoms interacting through the Lennard-Jones shifted-forceg potential truncated at a distance of 1.49 ao, where a. is the lattice constant at zero temperature. The potential parameters are chosen to represent copper, a = 2.31518, E = 0.167 eV. With this choice the T = OK lattice constant a. has the value 3.601 8 (experimental value is 3.60431) and the bulk melting temperature is 1250 K (experimental value is 1356 K). Integration of equations of motion is carried out over a time step size At = 1.84 x 10-15sec.

*Work supported by the U.S. Department of Energy,

BES-Materials Sciences, under Contract W-31-109-Eng-38.

&)permanent address: Dept. of Nuclear Engineering, MIT, Cambridge, MA 02139

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988544

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C5-382 JOURNAL DE PHYSIQUE

11. THE SEMI-PERIODIC BORDER

The b a s i c i d e a of t h e new b o r d e r t r e a t m e n t i s t o d i v i d e t h e s i m u l a t i o n c e l l ( s o - c a l l e d Region I ) i n t o a c e n t r a l i n t e r f a c e r e g i o n which c o n t a i n s t h e g r a i n boundary, bounded a t e i t h e r end by a b u f f e r r e g i o n ( s e e Fig. 1.). T h i s com- p u t a t i o n a l c e l l ( i n which Newton's e q u a t i o n s of motion a r e s o l v e d e x p l i c i t l y f o r e v e r y atom) i s t h e n s u r r o u n d e d by a s o - c a l l e d Region I1 which s i m u l a t e s t h e embedding of t h e i n t e r f a c e r e g i o n i n a b u l k i d e a l - c r y s t a l environment. To a c c o m p l i s h t h i s , t h e atom p o s i t i o n s i n Region 11 a r e chosen t o be a r e p l i c a of t h e atom p o s i t i o n s i n t h e c o r r e s p o n d i n g b u f f e r . The a d v a n t a g e s of t h i s Region I- Region I1 t r e a t m e n t a r e : ( 1 ) no e x t r a boundary needs t o be i n t r o d u c e d ; ( 2 ) t h e i n t e r f a c e r e g i o n may be e n l a r g e d d u r i n g s i m u l a t i o n w i t h o u t s i g n i f i c a n t l y per- t u r b i n g t h e b u f f e r r e g i o n , and ( 3 ) atoms i n t h e b u f f e r r e g i o n s behave a s i f i n a b u l k i d e a l - c r y s t a l environment, i . e . , t h e b u f f e r r e g i o n s a r e n o t h e l d a t T = OK a s i n t h e c a s e of t h e f i x e d b o r d e r . A d i s a d v a n t a g e i s t h a t s u r f a c e h e a t i n g o c c u r s i n t h e l a y e r s a d j a c e n t t o Region I1 s i n c e atoms moving i n t h e b o r d e r r e g i o n s w i l l do work on t h e atoms i n t h e b u f f e r r e g i o n s . T h i s h e a t i n g can be s u p p r e s s e d by t e m p e r a t u r e r e s c a l i n g .

To d e m o n s t r a t e t h a t t h i s b o r d e r t r e a t m e n t i s p h y s i c a l , we compare t h e v a r i a t i o n of t h e p o t e n t i a l e n e r g y a l o n g t h e grain-boundary p l a n e normal

( 2 - d i r e c t i o n ) f o r a n i d e a l c r y s t a l ( C = 1 ) w i t h and w i t h o u t t h e b o r d e r t r e a t m e n t , t h e l a t t e r c a s e b e i n g t h e u s u a l 3 d - p e r i o d i c b o r d e r . It c a n be s e e n i n Fig. 2 t h a t t h e p r e s e n c e of t h e b o r d e r i n t r o d u c e s no d i s c e r n i b l e changes i n t h e e n e r g y t h r o u g h o u t t h e c e l l . Other p r o p e r t i e s s u c h a s v i b r a t i o n a l a m p l i t u d e , p r e s s u r e and l a t t i c e e x p a n s i o n show t h e same b e h a v i o r . We t h e r e f o r e r e g a r d t h e s e r e s u l t s a s e v i d e n c e t h a t t h e b o r d e r t r e a t m e n t f o r m u l a t e d h e r e i s s a t i s f a c t o r y .

A

REGION I1

BUFFER 2

INTERFACE REGION

BUFFER 1

B i c r y s t a l s i m u l a t i o n model showing an i n t e r f a c e r e g i o n between two b u f f e r r e g i o n s ; t h e l a t t e r a r e s e m i - p e r i o d i - c a l l y r e p e a t e d i n t h e &z d i r e c t i o n s t o produce t h e b o r d e r r e g i o n s (Region 11) a s s o c i a t e d w i t h c r y s t a l s 1 and 2. 2 d - p e r i o d i c b o r d e r c o n d i t i o n s a r e imposed a l o n g t h e o t h e r two d i r e c t i o n s (x and y d i r e c t i o n s ) .

I REGION 31

2

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Distance (a )

Fig. 2. Potential energy distribution along the z-direction in a single crystal model with the present semi-periodic border treatment (open squares) and with the conventional 3d-periodic border (full squares).

111. (100) C = 29 TWIST BOUNDARY

A (100) E = 29, @ = 43.6O twist boundary is chosen for the present work. To follow the structural evolution of the two crystals as the temperature is raised,

Nn 2

we monitor the static structure factor S (K) _n

_-

= ( N ~ ) - ~ 17 exp(i5.r. )

1

^{layer by}

j=l ^-J

layer, where Nn is the number of atoms in layer n. The choice of the vec- 4 n

tor depends on the crystal orientation. Our choice of K is _-1

-

_[0,1,1l1

a

JT

4n [O,1,1] for the lower and upper crystals respectively. In the

and IC2 is

-

a

²

absence of thermal disorder, we find S ( K ) = 1 for any layer n in the lower n -1

crystal, and for those n in the upper crystal S (K ) has a quite low n -1

value, -0.01. Similarly Sn(K2) would have the same property with lower and upper crystals exchanged. By plotting S (K ) and Sn(K2) as a function of n one can

n -1

therefore readily identify those layers which have become thermally disordered, and the quantitative measure of the disorder allows a delineation of the spatial extent of the disordered region.

Since any observation of premelting behavior depends on knowing the bulk melting temperature, we first determine Tm by performing constant-stress simulations at various temperatures using a single-crystal model of 108 atoms

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JOURNAL DE PHYSIQUE

Temperature (K)

Fig. 3. Variation of potential energy (open squares) and lattice parameter (full squares) with temperature at constant zero stress in a single-crystal model of 108 atoms with 3d-periodic border. Data is averaged over 1000 tiroe steps after 2000 time steps of equilibration.

with the same potential function and 3d-periodic border conditions. The temperature variation of the potential energy and the lattice parameter are shown in Fig. 3. From this data we determine T, to lie between 1200 K and 1300 K.

Figure 4 shows the static structure-factor results for the (100) twist boundary at two temperatures. It can be seen that crystalline order is well maintained at 1000 K, but at 1100 K there is a substantial region of disorder at

the interface where S (K)

<

^0.1. The,indicated structural disordering occurs in n

-

planes parallel to the grain-boundary plane. To give a measure of the structural ordering in the direction perpendicular to these planes we show in Fig. 5 the distribution of atoms along the z-direction. The system clearly has a well- separated planar structure at 1000 K. At 1100 K atoms in the interface region have acquired sufficient mobility to cause some overlap between several adjacent planes.

- 1 2 - 8 - 4 0 4 8 1 2

Distance (a)

Fig. 4. Static structure factor S(K) for two & vectors, each chosen to be a reciprocal lattice vector in the corresponding half of the bicrystal.

Open squares are for 1000 K, full squares are for 1100 K.

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1 OOOK 11

OOK

6

.- i!?

F

0 -0

E

8 2 6

0

-

8 - 4 0 4 8 - 8

-

4 0 4 8

Distance (a) Distance (a)

Fig. 5. Atom number density across the bicrystal simulation cell showing a somewhat off centered interface region (due to grain boundary migration) and the buffer regions on either side. Shown are values averaged over 1000 time steps after a simulation time of 13000 time steps (1000 K) and 20000 time steps (1100 K), respectively.

IV. CONCWS IONS

We have presented a method of molecular dynamics simulation suitable for the study of bicrystals at elevated temperatures. The method incorporates the Parrinello-Rahman technique for simulation at constant stress with a new border treatment which avoids the introduction of a second grain boundary and the unphysical constraints of a rigid border. We find that our border formulation gives quite satisfactory results in describing the structural stability against thermal disordering of a (100) twist bicrystal of Lennard-Jones particles.

At T/Tm

-

0.9 there is some indication of a structural instability in the grain- boundary region; however longer simulations are necessary to reach definite conclusions.

ACKNOWLEDGRMENTS: The authors wish to acknowledge a grant of computer time on the Energy Research CRAY XMP at the Magnetic Fusion Computational Center at Livermore. This work was supported by the U.S. Department of Energy,

BES-Materials Sciences, under Contract W-31-109-Eng-38. One of us (TN) would like to acknowledge the support of a U.S. Department of Energy fellowship in Radioactive Waste Management. We have greatly benefited from many fruitful discussions with J. Lutsko.

1. T. Nguyen, P. S. Ho, T. Kwok, C. Nitta, and S. Yip, Phys. Rev. Letters,=

1919 (1986).

2. G. Cicotti, M. Guillope and V. Pontikis, Surf. Sci.,

144,

67 (1984);

M. Guillope, J. de Physique,% 1347 (1986).

3. D. Wolf, J. de Physique Colloque C4, ~01.46, C4-197 (1985).

4. G. H. Bishop, R. J. Harrision, T. Kwok, and S. Yip, in Progress in Material Science, J. W. Christian, P. Haasen, and T. B. Massalski, eds. (Pergamon, Oxford. 1972). - . D.

-

49.

5. T. Kwok, P. S. Ho, and S. Yip, Phys. Rev. Bx,5354 (1984).

6. M. Parrinello and A. Rahman, J. Appl. Phys.52, 7182 (1981).

7. J. J. Nicolas, K. E. Gubbins, W. B. Street, and D. J. Tildesley, Mol. Phys.

37, 1429 (1979).