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Evolution of the antiphase domain boundary upon interdiffusion
Physics Letters A 163 (1992) 127-129 Nonh-HoUand
PHYSICS LETTERS A
Evolution of the antiphase domain boundary upon interdiffusion V.B. S a p o z h n i k o v a n d M . G . G o l d i n e r Centerfor Scientific Research, Investtgation Automation and Metrology, Moldavian Academy of Sciences, Grosul Street 3/2, Kishinev 277028, Moldavia Received 24 October 1991;accepted for publication 16 December 199 l Communicated by V.M. Agranovich
The evolution of the antiphase domain boundary upon interdiffusion on a square lattice is investigated by computer simulation. It is found that the boundary is a self-affine fractal and that its evolution can be described by a double scaling. For the boundary width the relationship a(t, L) =L~f(t/L "/p) is valid, a being the boundary width, t the diffusion time and L the lattice length. The exponents are a,=0.57+_0.02, fl~=4I. The same scaling is found for the boundary energy (length) with exponents or/= 1.00_+0.02, fit= 0.10_+0.01.
In recent years surfaces of aggregates formed by stochastic processes, such as deposition and growth, were investigated both by computer simulation and by theoretical methods (see, for example, references in ref. [ 1 ] ). For such aggregates the dependence of the surface width tr upon time t and upon substrate length L can be written in a double scaling form [ 2 ], a(t, L ) = L " f ( t / L "/~) ,
( 1)
where f ( x ) o c x p (aoct ~) as x ~ 0 and f ( x ) = c o n s t (trocL ~) as x ~ o o . Kardar et al. [ 3 ] proposed the following equation of Langevin type for surface growth: Oh(x, t) _ vVEh(x ' t ) + ½2[Vh(x, t ) ] 2 + r/(x, t) Ot
(2) where h(x, t) measures the height o f the surface relative to the average height. The first term describes the boundary relaxation by surface tension, the second (nonlinear) term is caused by the slope dependence of the phase growth rate and the third term is white noise. From this equation Kardar et al. [ 3 ] obtained the exponents in ( 1 ). For a 2D system ( 1D boundary) ot = ½, fl= ~ if 2 = 0. Previously Edwards and Wilkinson [4] derived the linear Langevin equation (2 = 0) for the boundary which yielded the exponents ot = ½, fl= ~.
Recently we investigated the evolution of the diffusion couple in the cases o f zero [5,6 ], positive [ 7 ] and negative [ 8 ] mixing energy o f the components. In the case of positive mixing energy it is thermodynamically justified for the system to be separated into two phases by an interface [ 7 ]. The interface evolves into a self-affine fractal upon interdiffusion and its evolution corresponds to the predictions o f the linear version o f the Langevin equation. In the case of negative mixing energy [ 8 ], the ordered phase AB is formed in the diffusion zone having a square unit cell with two atoms A on one diagonal and two atoms B on the other. In contrast to the case of positive mixing energy the interface is a self-similar fractal. In the course of the growth o f the phase AB, antiphase domains of two types were formed depending on which o f the two sublattices was occupied by atoms A or B. However, the evolution o f the domains was not studied in ref. [ 8 ]. Here we investigated the geometry and the laws governing the evolution of the antiphase domain boundaries upon interdiffusion. These problems are interesting both in themselves and because the evolution o f the boundaries governs the domain growth. The atoms were placed on a square lattice. Originally, they formed two antiphase domains separated by a straight line joining middle points o f the lattice lateral sides. Periodic boundary conditions were im-
0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
127
Volume 163, number 1,2
PHYSICS LETTERS A
posed on the lattice. A vacancy m o v e d over the lattice and j u m p e d each t i m e to one o f the four neighb o u t sites thus carrying out the interdiffusion. The activation energy o f the vacancy j u m p was taken to be E = E o + ½n~, where n is the change in the n u m b e r o f heterogeneous b o u n d s in the sample as a result o f the j u m p and e=¢AB--0.5(~AA-I"~BB)is the mixing energy o f the a t o m s A and B (eAA, (AB and eBB are the binding energies for the corresponding pairs). Then the difference between the activation energies o f the direct j u m p and the reverse one is just nu. This means that the relative probability o f a diffusion j u m p to a given site is p r o p o r t i o n a l to e x p ( - ½nu) where u = e / k T is the reduced mixing energy. The simulation was performed at u = - 1. The results were averaged over 2 0 - 2 0 0 runs. In the course o f diffusion the antiphase d o m a i n b o u n d a r y roughens (fig. 1 ). We investigated the kinetics o f the growth o f its mean width cr ( d o u b l e mean square deviation from the central position) and
4
10
vacancy jumps
r
]
(a 6 10 vacancy jumps
I
Fig. 1. Antiphase domains on a 256 × 128 lattice at different moments. The brackets show the boundary mean widths. 128
9 March 1992
its length (energy) / (fig. 2 ). First, they diverge with time as a power law with exponents fl~-~0.24+0.01 and fl~_~0.10 _+0.01. Then the b o u n d a r y width and length saturate to constant values c r ( ~ , L ) and l(~c, L ) , respectively. These limiting values a p p e a r e d to d e p e n d upon the lattice length as a power law with exponents c ~ = 0.57 + 0.02 and c~/= 1.00_+ 0.02. The relations o b t a i n e d can be written in the double scaling form ( 1 ). The geometry of the b o u n d a r y was also described by the height difference correlation function: C ( A r ) = ( [ h ( x ) - h ( x + A x 9 ]2)1/2. As is often done [9] the b o u n d a r y was replaced by the single valued function h (x) in such a way that only the largest value o f the surface height was considered at each coordinate x. Up to a certain inner cut-off length Axm, the correlation function turned out to have a power law form C ( & x ' ) - ~ ( & r ) " , with the exponent H = 0 . 5 1 _+0.02. That means that the b o u n d a r y is a self-affine fractal and that Arm is its correlation length in the direction o f the original boundary. The correlation length increases with time. W h e n it reaches the lattice length L, the b o u n d a r y width stops growing. As was shown in ref. [9] 11 must be equal to ~e~. The difference between H and c~ found and the difference between c~ and the value ½ predicted by the Langevin equation are probably due to the fact that the b o u n d a r y is not a single valued function o f the coordinate. One can suppose that in such cases the prediction is valid that H = ½ because the b o u n d a r y is replaced by a single valued function for the calculation of the exponent tt. The published results for surface evolution fall into two universality classes which are consistent with the predictions o f either the linear ( 2 = 0 ) or the nonlinear version o f (2). The values o f the exponents obtained by us (c~_~ ½, fl~-~ ¼) suggest that in contrast to the majority of the deposition and growth models, the system under study belongs to the first universality class corresponding to )~= 0 (taking into account the above remark that the b o u n d a r y is not a single valued function), that is, to the same universality class as the model of the diffusive evolution o f the interface at positive mixing energy [7 ]. This result can be u n d e r s t o o d taking into account that in the investigated system the coefficient 2 o f the nonlinear term in ( 2 ) is zero because it is p r o p o r t i o n a l
Volume 163, number 1,2
PHYSICS LETTERS A
9 March 1992 1000 L=256
1=256
~
j ~ " j/J"
L=128
_
~
~
L= 128
f o
~f~_
o
L=32
100
_._...-.~--"--,,~ L = 6 4
l j L=32
o
-fi
o
~
/
ill
i piHl.i
0 -~
i j
i~lllll
i iiHml
i t IHl.I
iii
I lllHId
i iii,i.~
...,~.,.~ L = 1 6
i i~i~,~I
, ,,.,,,I
,,i~i.iI
i lJllllll
1010-~ 1 10 102 103 1 0 ' v a c a n c y j u m p s per site
i i0 i0 2 10 3 10 ' v a c a n c y j u m p s per site
Fig. 2. Boundary evolution for different lattice lengths.
to the phase growth rate [ 1 ] while the b o u n d a r y m e a n position remains constant. The results obtained allow the following conclusions: - In contrast to the interface between the ordered phase and the disordered one the antiphase d o m a i n b o u n d a r y evolves into a self-affine fractal u p o n interdiffusion. - Its evolution is described by scaling relation ( l ) where a o = 0.57 _ 0.02 a n d ft,= ~. - The same equation is valid for the b o u n d a r y length (energy), the exponents being a t = 1.00 + 0.02, ill=0.10_+ 0.01. - The interface evolution is consistent with the predictions of the linear version of the Langevin equation (taking into account the above remark that the b o u n d a r y is not a single valued f u n c t i o n ) . - The antiphase d o m a i n b o u n d a r y belongs to the
same universality class as the interface between phases A a n d B at positive mixing energy [ 7 ].
References
[ 1] J. Kung, J. Phys. A 22 (1989) L769. [2 ] F. Family and T. Vices,J. Phys. A 18 ( 1985) L75. [3] M. Kardar, G. Parisi and Y.C. Zhang, Phys. Rev. Lett. 56 (1986) 889. [4] S.F. Edwards and D.R. Wilkinson, Proc. R. Soc. A 381 (1982) 17. [5] V.B. Sapozhnikov and M.G. Goldiner, Sov. Phys. JETP 67 (1988) 177. [6] V.B. Sapozhnikovand M.G. Goldiner, J. Phys. A 23 (1990) 5309. [7] V.B. Sapozhnikov and M.G. Goldiner, Phys. Lett. A 162 (1992) 59. [ 8 ] V.B. Sapozhnikovand M.G. Goldiner, J. Phys. A 24 ( 1991 ) L853. [9] P. Meakin, Physica D 38 (1989) 252.
129
PHYSICS LETTERS A
Evolution of the antiphase domain boundary upon interdiffusion V.B. S a p o z h n i k o v a n d M . G . G o l d i n e r Centerfor Scientific Research, Investtgation Automation and Metrology, Moldavian Academy of Sciences, Grosul Street 3/2, Kishinev 277028, Moldavia Received 24 October 1991;accepted for publication 16 December 199 l Communicated by V.M. Agranovich
The evolution of the antiphase domain boundary upon interdiffusion on a square lattice is investigated by computer simulation. It is found that the boundary is a self-affine fractal and that its evolution can be described by a double scaling. For the boundary width the relationship a(t, L) =L~f(t/L "/p) is valid, a being the boundary width, t the diffusion time and L the lattice length. The exponents are a,=0.57+_0.02, fl~=4I. The same scaling is found for the boundary energy (length) with exponents or/= 1.00_+0.02, fit= 0.10_+0.01.
In recent years surfaces of aggregates formed by stochastic processes, such as deposition and growth, were investigated both by computer simulation and by theoretical methods (see, for example, references in ref. [ 1 ] ). For such aggregates the dependence of the surface width tr upon time t and upon substrate length L can be written in a double scaling form [ 2 ], a(t, L ) = L " f ( t / L "/~) ,
( 1)
where f ( x ) o c x p (aoct ~) as x ~ 0 and f ( x ) = c o n s t (trocL ~) as x ~ o o . Kardar et al. [ 3 ] proposed the following equation of Langevin type for surface growth: Oh(x, t) _ vVEh(x ' t ) + ½2[Vh(x, t ) ] 2 + r/(x, t) Ot
(2) where h(x, t) measures the height o f the surface relative to the average height. The first term describes the boundary relaxation by surface tension, the second (nonlinear) term is caused by the slope dependence of the phase growth rate and the third term is white noise. From this equation Kardar et al. [ 3 ] obtained the exponents in ( 1 ). For a 2D system ( 1D boundary) ot = ½, fl= ~ if 2 = 0. Previously Edwards and Wilkinson [4] derived the linear Langevin equation (2 = 0) for the boundary which yielded the exponents ot = ½, fl= ~.
Recently we investigated the evolution of the diffusion couple in the cases o f zero [5,6 ], positive [ 7 ] and negative [ 8 ] mixing energy o f the components. In the case of positive mixing energy it is thermodynamically justified for the system to be separated into two phases by an interface [ 7 ]. The interface evolves into a self-affine fractal upon interdiffusion and its evolution corresponds to the predictions o f the linear version o f the Langevin equation. In the case of negative mixing energy [ 8 ], the ordered phase AB is formed in the diffusion zone having a square unit cell with two atoms A on one diagonal and two atoms B on the other. In contrast to the case of positive mixing energy the interface is a self-similar fractal. In the course of the growth o f the phase AB, antiphase domains of two types were formed depending on which o f the two sublattices was occupied by atoms A or B. However, the evolution o f the domains was not studied in ref. [ 8 ]. Here we investigated the geometry and the laws governing the evolution of the antiphase domain boundaries upon interdiffusion. These problems are interesting both in themselves and because the evolution o f the boundaries governs the domain growth. The atoms were placed on a square lattice. Originally, they formed two antiphase domains separated by a straight line joining middle points o f the lattice lateral sides. Periodic boundary conditions were im-
0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
127
Volume 163, number 1,2
PHYSICS LETTERS A
posed on the lattice. A vacancy m o v e d over the lattice and j u m p e d each t i m e to one o f the four neighb o u t sites thus carrying out the interdiffusion. The activation energy o f the vacancy j u m p was taken to be E = E o + ½n~, where n is the change in the n u m b e r o f heterogeneous b o u n d s in the sample as a result o f the j u m p and e=¢AB--0.5(~AA-I"~BB)is the mixing energy o f the a t o m s A and B (eAA, (AB and eBB are the binding energies for the corresponding pairs). Then the difference between the activation energies o f the direct j u m p and the reverse one is just nu. This means that the relative probability o f a diffusion j u m p to a given site is p r o p o r t i o n a l to e x p ( - ½nu) where u = e / k T is the reduced mixing energy. The simulation was performed at u = - 1. The results were averaged over 2 0 - 2 0 0 runs. In the course o f diffusion the antiphase d o m a i n b o u n d a r y roughens (fig. 1 ). We investigated the kinetics o f the growth o f its mean width cr ( d o u b l e mean square deviation from the central position) and
4
10
vacancy jumps
r
]
(a 6 10 vacancy jumps
I
Fig. 1. Antiphase domains on a 256 × 128 lattice at different moments. The brackets show the boundary mean widths. 128
9 March 1992
its length (energy) / (fig. 2 ). First, they diverge with time as a power law with exponents fl~-~0.24+0.01 and fl~_~0.10 _+0.01. Then the b o u n d a r y width and length saturate to constant values c r ( ~ , L ) and l(~c, L ) , respectively. These limiting values a p p e a r e d to d e p e n d upon the lattice length as a power law with exponents c ~ = 0.57 + 0.02 and c~/= 1.00_+ 0.02. The relations o b t a i n e d can be written in the double scaling form ( 1 ). The geometry of the b o u n d a r y was also described by the height difference correlation function: C ( A r ) = ( [ h ( x ) - h ( x + A x 9 ]2)1/2. As is often done [9] the b o u n d a r y was replaced by the single valued function h (x) in such a way that only the largest value o f the surface height was considered at each coordinate x. Up to a certain inner cut-off length Axm, the correlation function turned out to have a power law form C ( & x ' ) - ~ ( & r ) " , with the exponent H = 0 . 5 1 _+0.02. That means that the b o u n d a r y is a self-affine fractal and that Arm is its correlation length in the direction o f the original boundary. The correlation length increases with time. W h e n it reaches the lattice length L, the b o u n d a r y width stops growing. As was shown in ref. [9] 11 must be equal to ~e~. The difference between H and c~ found and the difference between c~ and the value ½ predicted by the Langevin equation are probably due to the fact that the b o u n d a r y is not a single valued function o f the coordinate. One can suppose that in such cases the prediction is valid that H = ½ because the b o u n d a r y is replaced by a single valued function for the calculation of the exponent tt. The published results for surface evolution fall into two universality classes which are consistent with the predictions o f either the linear ( 2 = 0 ) or the nonlinear version o f (2). The values o f the exponents obtained by us (c~_~ ½, fl~-~ ¼) suggest that in contrast to the majority of the deposition and growth models, the system under study belongs to the first universality class corresponding to )~= 0 (taking into account the above remark that the b o u n d a r y is not a single valued function), that is, to the same universality class as the model of the diffusive evolution o f the interface at positive mixing energy [7 ]. This result can be u n d e r s t o o d taking into account that in the investigated system the coefficient 2 o f the nonlinear term in ( 2 ) is zero because it is p r o p o r t i o n a l
Volume 163, number 1,2
PHYSICS LETTERS A
9 March 1992 1000 L=256
1=256
~
j ~ " j/J"
L=128
_
~
~
L= 128
f o
~f~_
o
L=32
100
_._...-.~--"--,,~ L = 6 4
l j L=32
o
-fi
o
~
/
ill
i piHl.i
0 -~
i j
i~lllll
i iiHml
i t IHl.I
iii
I lllHId
i iii,i.~
...,~.,.~ L = 1 6
i i~i~,~I
, ,,.,,,I
,,i~i.iI
i lJllllll
1010-~ 1 10 102 103 1 0 ' v a c a n c y j u m p s per site
i i0 i0 2 10 3 10 ' v a c a n c y j u m p s per site
Fig. 2. Boundary evolution for different lattice lengths.
to the phase growth rate [ 1 ] while the b o u n d a r y m e a n position remains constant. The results obtained allow the following conclusions: - In contrast to the interface between the ordered phase and the disordered one the antiphase d o m a i n b o u n d a r y evolves into a self-affine fractal u p o n interdiffusion. - Its evolution is described by scaling relation ( l ) where a o = 0.57 _ 0.02 a n d ft,= ~. - The same equation is valid for the b o u n d a r y length (energy), the exponents being a t = 1.00 + 0.02, ill=0.10_+ 0.01. - The interface evolution is consistent with the predictions of the linear version of the Langevin equation (taking into account the above remark that the b o u n d a r y is not a single valued f u n c t i o n ) . - The antiphase d o m a i n b o u n d a r y belongs to the
same universality class as the interface between phases A a n d B at positive mixing energy [ 7 ].
References
[ 1] J. Kung, J. Phys. A 22 (1989) L769. [2 ] F. Family and T. Vices,J. Phys. A 18 ( 1985) L75. [3] M. Kardar, G. Parisi and Y.C. Zhang, Phys. Rev. Lett. 56 (1986) 889. [4] S.F. Edwards and D.R. Wilkinson, Proc. R. Soc. A 381 (1982) 17. [5] V.B. Sapozhnikov and M.G. Goldiner, Sov. Phys. JETP 67 (1988) 177. [6] V.B. Sapozhnikovand M.G. Goldiner, J. Phys. A 23 (1990) 5309. [7] V.B. Sapozhnikov and M.G. Goldiner, Phys. Lett. A 162 (1992) 59. [ 8 ] V.B. Sapozhnikovand M.G. Goldiner, J. Phys. A 24 ( 1991 ) L853. [9] P. Meakin, Physica D 38 (1989) 252.
129