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Temperature dependence of domain growth kinetics in two dimensions
Journal of Magnetism and Magnetic Materials 157/158 (1996) 345-346
~iilPiJournalof magnetism ~Hand magnetic ~ H materials
ELSEVIER
Temperature dependence of domain growth kinetics in two dimensions C. Mor6n *, M. Mora Dpto. S.LA., E.U. Informdtica (U.P.M.), 28031 Madrid, Spain Abstract W e have carried out Monte Carlo simulations to study the effect of temperature on the growth kinetics of a circular grain. As predicted by d = 3 theories of domain kinetics, the circular domain shrinks linearly with time as A(t) = A o - at, where A 0 and A(t) are the initial and instantaneous areas, respectively.
Keywords: Domain growth; Domain dynamics; Monte Carlo simulation
In this paper we report additional computer simulation results of a ferromagnetic Q-state Potts model with Glanber dynamic which is rapidly quenched from T >> Tc to T = 0. W e consider the Hamiltonian,
H = - J E 8sisF NN
(1)
where S i is the Q state of the spin on site i (1 < S / < Q) and 8sis~ is the Kronecker delta function. The sum is over nearest-neighbor spins, Using Monte Carlo techniques [1], we have studied different model domain geometries with well-defined initial conditions as well as systems which are quenched from the disordered state (T>> Tc ) to the ordered state (T < Tc). To reduce the effects of the boundary, we have studied very large systems, up to 200 × 200 sites with periodic boundary conditions. The dynamics of the spin are introduced in the standard manner for the Glauber dynamic. We choose a site at random and compute the change in energy A E required to flip the spin at that site. W e next obtain the transition probability Wa using
[e-~E/k~ ~
W ° = ( 1,
'
AE>0, AE<0,
Fig. 1 shows the instantaneous spin configurations for the Q = 6 Potts model on the square lattices that were quenched from high temperatures (T >> Tc ) to T-~ 0. In analyzing the results of the our simulations, we have applied the generalized form of Euler's equation [2,3] to compute the mean area per grain ( A ( t ) ) . This formula gives the total number of domains D(t) in terms of the number of edges E(t) and vertices V(t) as
D(t)
-
E ( t ) + V ( t ) = f O,
for an infinite system, for a finite system.
(3)
(2)
where k B is the Boltzmann constant and T is temperature. Using Eq. (1), the transition probability We is computed and compared with a random number R (0 _< R _< 1) and the spin is changed if We > R. In this way, the configurational averaging is obtained by averaging the data over many runs and we define the unit of time as 1 Monte Carlo step per spin ( M C S / S ) .
* Corresponding author. Fax: ÷ 34-1-336-7522.
1,
MCS = 80
MCS =240
MCS=4SO
MCS=$O0
Fig. 1. Spin configurations for the Q = 6 Potts model on the square lattice that were quenched from T >> Tc to T - 0.
0304-8853/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. SSDI 0 3 0 4 - 8 8 5 3 ( 9 5 ) 0 1 147-1
C. Mordn, M. Mora/ Journal of Magnetism and Magnetic Materials 157/158 (1996)345-346
346
Further, for an infinite system, or one in which periodic boundary conditions are maintained, every edge is twoended and on a triangular lattice each vertex is three-rayed, yielding in this case 2 E ( t ) = 3V(t), and so D(t) = V(t)/2. Thus the mean area on the triangular lattice is related to the total area A T by (A(t)) = 2AT/V(t). This mean area ( A ( t ) ) was calculated in Ref. [2], using this formula for both the triangular and square lattices. In our case, we have counted the number of three-rayed vertices as a function of time for the Q = 6 state Potts model on a 200 x 200 lattice (Fig. 2). The time dependence of the mean domain size can be approximated as R = Ct'; this was determined by monitoring the number of three-domain vertices during the simulation. W e have calculated the exponent n as a function of Q for the triangular lattice at T = 0. The exponent n is found to decrease linearly with increasing Q, for Q < 30. Extrapolating to the Ising limit (Q = 2), we find that the well-known result ( n = 1 / 2 ) is reproduced. For Q > 30, the exponent n is independent of Q and has a value of 0.41. However, in the case of the standard transcription from a lattice gas to an antiferromagnet, the spins are described by a nearest-neighbour Ising Hamiltonian
H = - J E SiSj.
(4)
ij Using the Glauber dynamic, if the initial configuration is a circular domain of one sublattice surrounded by a sea of another degenerate sublattice on an 200 X 200 square lattice (Fig. 3), the average area of the domain is then obtained by averaging the data over at least 20 configurations for T < 0.6 Tc and 40 configurations for T > 0.6 Tc. 600
Itlltllllllllllllllllllllllllllllllllllllll
"<:400' I,I Z 30O'
• MCS = 0
Ii
tP,
MC$ = 200
"
.
O'"
M12S=400
MCS=600
Fig. 3. Evolution of an initially circular domain
for various
instants of time for T = 0.6 T c .
Fig. 3 shows snapshots of the evolution of an initially circular domain at different times for T = 0.6 Tc. For the purposes of this display, every alternate spin in a given configuration has been flipped, which results in ferromagnetic domains of up spins (circles) and down spins (white region). It is quite obvious that the circular domain roughens very rapidly because of thermal fluctuations. For T _< 0.5 Tc, we found that the circular domain remained essentially circular throughout the evolution, whereas for T _ 0.8 Tc, thermal fluctuations are so strong that the domain becomes extremely wavy in less than 10 MC steps. In this way, it is found that the instantaneous average area A(t) decreases linearly with time, A ( t ) = A o - at, where A o is the initial area of the domain. For T < 0.6 Tc, the slope a is constant, as expected from Allen and Cahn [4]. However, for T > 0.6 Tc, a decreases linearly with increasing temperature. The physical origin of this decreasing rate of shrinking is the roughening of the domain boundary to a noncircular shape by the thermal fluctuations, whose effects are important in two dimensions [511 In conclusion, the roughening of the domain boundary by thermal fluctuations gives rise to a strong temperature dependence of the rate of domain growth. Good agreement is found when the MC results are compared with the roughening model. References
200.
100
"'~'66 . . .zoo . . . . . . .zoo . . . . . . '. 4. .~. .'. . .s. & ......6~
700 s00
TIME (MCS/spin)
goo
Fig. 2, Growth rate of the mean area of a domain, (A(t)) for the Q = 6 state Potts model on a 200 × 200 lattice at T = 0.1 Tc .
[1] C. Mordn and M. Mora, J. Magn. Magn. Mater. 133 (1994) 57. [2] P.S. Sahni, D.J. Srolovitz, G.S. Grest, M.P. Anderson and S.A. Safran, Phys. Rev. B 28 (1983) 2705. [3] M.P. Anderson, D.J. Srolovitz, G.S. Grest and P.S. Sahni, Acta Metall. 32 (1984) 783. [4] S.M. Allen and J.W. Cahn, Acta Metall. 27 (1979) 1085. [5] P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49 (1974) 135.
~iilPiJournalof magnetism ~Hand magnetic ~ H materials
ELSEVIER
Temperature dependence of domain growth kinetics in two dimensions C. Mor6n *, M. Mora Dpto. S.LA., E.U. Informdtica (U.P.M.), 28031 Madrid, Spain Abstract W e have carried out Monte Carlo simulations to study the effect of temperature on the growth kinetics of a circular grain. As predicted by d = 3 theories of domain kinetics, the circular domain shrinks linearly with time as A(t) = A o - at, where A 0 and A(t) are the initial and instantaneous areas, respectively.
Keywords: Domain growth; Domain dynamics; Monte Carlo simulation
In this paper we report additional computer simulation results of a ferromagnetic Q-state Potts model with Glanber dynamic which is rapidly quenched from T >> Tc to T = 0. W e consider the Hamiltonian,
H = - J E 8sisF NN
(1)
where S i is the Q state of the spin on site i (1 < S / < Q) and 8sis~ is the Kronecker delta function. The sum is over nearest-neighbor spins, Using Monte Carlo techniques [1], we have studied different model domain geometries with well-defined initial conditions as well as systems which are quenched from the disordered state (T>> Tc ) to the ordered state (T < Tc). To reduce the effects of the boundary, we have studied very large systems, up to 200 × 200 sites with periodic boundary conditions. The dynamics of the spin are introduced in the standard manner for the Glauber dynamic. We choose a site at random and compute the change in energy A E required to flip the spin at that site. W e next obtain the transition probability Wa using
[e-~E/k~ ~
W ° = ( 1,
'
AE>0, AE<0,
Fig. 1 shows the instantaneous spin configurations for the Q = 6 Potts model on the square lattices that were quenched from high temperatures (T >> Tc ) to T-~ 0. In analyzing the results of the our simulations, we have applied the generalized form of Euler's equation [2,3] to compute the mean area per grain ( A ( t ) ) . This formula gives the total number of domains D(t) in terms of the number of edges E(t) and vertices V(t) as
D(t)
-
E ( t ) + V ( t ) = f O,
for an infinite system, for a finite system.
(3)
(2)
where k B is the Boltzmann constant and T is temperature. Using Eq. (1), the transition probability We is computed and compared with a random number R (0 _< R _< 1) and the spin is changed if We > R. In this way, the configurational averaging is obtained by averaging the data over many runs and we define the unit of time as 1 Monte Carlo step per spin ( M C S / S ) .
* Corresponding author. Fax: ÷ 34-1-336-7522.
1,
MCS = 80
MCS =240
MCS=4SO
MCS=$O0
Fig. 1. Spin configurations for the Q = 6 Potts model on the square lattice that were quenched from T >> Tc to T - 0.
0304-8853/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. SSDI 0 3 0 4 - 8 8 5 3 ( 9 5 ) 0 1 147-1
C. Mordn, M. Mora/ Journal of Magnetism and Magnetic Materials 157/158 (1996)345-346
346
Further, for an infinite system, or one in which periodic boundary conditions are maintained, every edge is twoended and on a triangular lattice each vertex is three-rayed, yielding in this case 2 E ( t ) = 3V(t), and so D(t) = V(t)/2. Thus the mean area on the triangular lattice is related to the total area A T by (A(t)) = 2AT/V(t). This mean area ( A ( t ) ) was calculated in Ref. [2], using this formula for both the triangular and square lattices. In our case, we have counted the number of three-rayed vertices as a function of time for the Q = 6 state Potts model on a 200 x 200 lattice (Fig. 2). The time dependence of the mean domain size can be approximated as R = Ct'; this was determined by monitoring the number of three-domain vertices during the simulation. W e have calculated the exponent n as a function of Q for the triangular lattice at T = 0. The exponent n is found to decrease linearly with increasing Q, for Q < 30. Extrapolating to the Ising limit (Q = 2), we find that the well-known result ( n = 1 / 2 ) is reproduced. For Q > 30, the exponent n is independent of Q and has a value of 0.41. However, in the case of the standard transcription from a lattice gas to an antiferromagnet, the spins are described by a nearest-neighbour Ising Hamiltonian
H = - J E SiSj.
(4)
ij Using the Glauber dynamic, if the initial configuration is a circular domain of one sublattice surrounded by a sea of another degenerate sublattice on an 200 X 200 square lattice (Fig. 3), the average area of the domain is then obtained by averaging the data over at least 20 configurations for T < 0.6 Tc and 40 configurations for T > 0.6 Tc. 600
Itlltllllllllllllllllllllllllllllllllllllll
"<:400' I,I Z 30O'
• MCS = 0
Ii
tP,
MC$ = 200
"
.
O'"
M12S=400
MCS=600
Fig. 3. Evolution of an initially circular domain
for various
instants of time for T = 0.6 T c .
Fig. 3 shows snapshots of the evolution of an initially circular domain at different times for T = 0.6 Tc. For the purposes of this display, every alternate spin in a given configuration has been flipped, which results in ferromagnetic domains of up spins (circles) and down spins (white region). It is quite obvious that the circular domain roughens very rapidly because of thermal fluctuations. For T _< 0.5 Tc, we found that the circular domain remained essentially circular throughout the evolution, whereas for T _ 0.8 Tc, thermal fluctuations are so strong that the domain becomes extremely wavy in less than 10 MC steps. In this way, it is found that the instantaneous average area A(t) decreases linearly with time, A ( t ) = A o - at, where A o is the initial area of the domain. For T < 0.6 Tc, the slope a is constant, as expected from Allen and Cahn [4]. However, for T > 0.6 Tc, a decreases linearly with increasing temperature. The physical origin of this decreasing rate of shrinking is the roughening of the domain boundary to a noncircular shape by the thermal fluctuations, whose effects are important in two dimensions [511 In conclusion, the roughening of the domain boundary by thermal fluctuations gives rise to a strong temperature dependence of the rate of domain growth. Good agreement is found when the MC results are compared with the roughening model. References
200.
100
"'~'66 . . .zoo . . . . . . .zoo . . . . . . '. 4. .~. .'. . .s. & ......6~
700 s00
TIME (MCS/spin)
goo
Fig. 2, Growth rate of the mean area of a domain, (A(t)) for the Q = 6 state Potts model on a 200 × 200 lattice at T = 0.1 Tc .
[1] C. Mordn and M. Mora, J. Magn. Magn. Mater. 133 (1994) 57. [2] P.S. Sahni, D.J. Srolovitz, G.S. Grest, M.P. Anderson and S.A. Safran, Phys. Rev. B 28 (1983) 2705. [3] M.P. Anderson, D.J. Srolovitz, G.S. Grest and P.S. Sahni, Acta Metall. 32 (1984) 783. [4] S.M. Allen and J.W. Cahn, Acta Metall. 27 (1979) 1085. [5] P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49 (1974) 135.