- Email: [email protected]
A new model of DLA under high magnetic field
COMPUTATIONAL MATERIALS SCIENCE
ELSEVIER
Computational
Materials
Science
10 (1998) 4650
A new model of DLA under high magnetic field Hiroshi Mizuseki *, Kazumi Tanaka, Keiko Kikuchi, Kaoru Ohno, Yoshiyuki Kawazoe Institute jix Materials Research, Tohoku Universi@ Sendai 980-77, Japan
Abstract A new Monte Carlo model is introduced to describe diffusion-limited aggregation (DLA) with extra forces arising from Lorentz’s and/or Coulomb forces. Furthermore, we simulate a behavior of multiparticle diffusive aggregation to examine the resultant pattern of crystal in electrochemical deposition. Different patterns grown under various external forces are produced by Monte Carlo simulations. In the present model, the basic movement of particles is a random walk, with different transition probabilities in different directions, which characterizes stochastically the effect of extra forces. In case of assuming a high magnetic field, pattern formations which are qualitatively different from the standard DLA model are observed and they are successfully compared with preexisting experiments (Mogi et al., 1991). The present numerical results of electrochemical deposition show that the generated patterns strongly depend on the force acting on ions and their concentration (Sawada et al., 1986). Copyright 0 1998 Elsevier Science B.V. Keywords: Fractal dimension; deposition;
Diffusion-limited Monte Carlo method
aggregation
(DLA); Magnetic
1. Introduction The phenomena system
of crystal
have recently
growth
attracted
tion [ 1,2]. Pattern-formation
in diffusive
considerable
examples
atten-
in diffusive
system include electrochemical deposition, crystal growth, viscous fingering, dielectric breakdown, chemical dissolution and bacterial colonies. An approximation to these phenomena is provided by the Laplacian growth model that can be simulated by the diffusion-limited aggregation (DLA) [ 11. These patterns generated by DLA are analyzed by computational and experimental methods. Several extensions of the DLA model have been proposed to take into
* Corresponding author. Tel.: f81 22 215 2054; fax: +8 1 22 215 2052; e-mail: [email protected]. 0927-0256/98/$19.00
Copyright
PII SO927-0256(97)00142-O
0 1998 Elsevier
Science
field; Crystal
growth;
Random
walk; Electrochemical
account electrochemical deposition [3], magnetic field [4], concentration [5], drift [6], convection [7] and sticking probability [5]. A variety of computer simulations relationship
have been carried out to investigate the between the patterns of the crystal and
the movement behavior of the particles. Consequently, the morphology of the crystal strongly depends on the dynamics of the crystal growth process. As a new type of perturbation, Institute
applied a high magnetic
Mogi et al. at our field for the den-
dritic growth of silver [8], lead [9] and zinc [lo] in solutions
and obtained new specific patterns of metal
leaves. Under a high magnetic field, the branches bend slightly and the shape of the envelope becomes circular. These results imply that the particle diffusion in the medium is strongly affected by the magnetic field and this decisive difference in the particle movement
B.V. All rights reserved
H. Mizuseki et al. /Computational
changes the final pattern from the DLA result. Since the direction of bending is reversed, according to the reversal of magnetic field, the observed bending of the crystal is attributed to the cyclotron motion of the Ag ions. These experimental works have also shown that the effect of magnetic field is universal, i.e., independent of the details of the chemical properties of the solution. Therefore, we expect to reproduce this universality observed in the experiments via computer simulation. The first aim of this paper is to examine various patterns of crystal growths under an external force involving the stochastic treatment of the cyclotron motion. Sawada et al. [ 11,121 show that the patterns in electrochemical deposition depend on the concentration of the solution and applied voltage of electrode. This experiment shows that different pattern formations under various conditions such as an ion concentration and an applied voltage in the Zn from a thin layer of ZnSO4 solution appeared. It is worthwhile to examine the subject more closely. Therefore, we have to inquire, to some extent, into the force that is affected by other ions. In this paper, we present a new model for the electrodeposition process. The multiparticle DLA simulation is carried out to simulate this behavior of particle [ 11,121, because the particle is affected by Coulomb forces from the electrode as well as from the other ions. The present paper is organized as follows: A new growth model is introduced in the next section, which is affected by external force. A large computer simulation is performed and the obtained numerical results are given in Section 3. They are compared with the experimental observations also. Sections 4 and 5 are devoted, respectively, to discussion and conclusion.
2. Model and numerical method Our present model is basically a Monte Carlo simulation on a square lattice. The inclusion of external forces shows that the pattern formation is governed not only by the diffusion field, but also by the magnetic field, gradient of concentration, electric field, and so on. The effect of the magnetic field is included by
Materials Science 10 (1998) 46-50
47
simply changing the probability of the movement perpendicular to the Monte Carlo movement. We assume two environments in crystal growth: high magnetic field and gradient of ion concentration. Multiparticle aggregations have been simulated at various concentration f in the range of 0.05-0.4 and crystal sizes from 500 x 500 to 800 x 800. In the present simulation, when the particle arrives to contact with the crystal, it sticks permanently, and a sticking probability between the ion and the crystal is 1. Furthermore, we have not allowed that two ions occupy the same square sites. In this simulation, the area far from the electrode was held at initial concentration f. During simulations, f is fixed. This is essentially the model decribed previously [4] where it was used to study the effect of external force in crystal growth.
3. Results 3.1. High magnetic field Numerical simulations were performed based on the models described in the preceding section. lypical growth patterns obtained by the present research are indicated in Figs. 1 and 2 together with the experimental results [8]. The numbers on the axes show the mesh points and therefore the sizes of the model clusters. By the standard DLA growth model (Fig. l), the well-known pattern with the fractal dimension of 1.67 is obtained for the square lattice case. Fig. 2(b) shows the experimentally observed crystal pattern under high magnetic field [81. This figure bears resemblance to the present simulation from the view point of the bending of the pattern. The numerically obtained pattern in Fig. 2(a) is basically similar to the pattern of the experimental result. The numbers and the thickness of the branches in Figs. 2(a) and (b) are still different because of the small number of particles in simulation compared with experiment. If we can perform quantitatively larger scale simulations which are not possible at the moment, the number of branches increase and accordingly the branches will become thinner.
H. Mizuseki et al. /Computational Materials Science 10 (1998) 46-50
48
1000
1
I
I
500
0
-500
-y
30
I
I
I
-500
0
500
(a)
10011
fb)
Fig. 1. (a) Simulated pattern of the standard DLA model on square lattice. (b) Observed field. The square at the center of the metal leaf is a piece of copper metal [8].
pattern
of silver leaves without
magnetic
lOoF
l500
aI
-8
4
l-
I-
001D
0
(b) Fig. 2. (a) Simulated crystal growth with the effect of Lorentz force and gradient of electrolyte concentration, respectively. The conditions used are a = 1.O, 0 = IO’, and ionization tendency parameter b = 0.035. The notation of parameters was referred to [4]. (b) Observed pattern of silver leaves under 8T. The magnetic field was applied perpendicular to the plate [8].
H. Mizuseki et al. /Computational Materials Science 10 (1998) 46-50
49
600
100
200
400
300
500
600
100
200
300
400
500
Fig. 3. Multiparticle diffusive aggregation to point electrode from a concentration (a)
3.2. Electrochemical
I
,
I
i
I
-0.2-
-
k<
-0.4 -
--
f =
200
300
400
500
600
0.05, (b) f = 0.1 and (c) f = 0.2.
4. Discussion
deposition
Typical growth patterns obtained by the present research are indicated in Fig. 3. Figs. 3(a)-(c) show the results of aggregation from a point for different f. Naturally, in low concentration, the obtained pattern is very similar to the standard DLA pattern. On the other hand, in high concentration, the number of branches increases and accordingly the space between the branches becomes small. In the case of random structures such as present study, we need the so-called densitydensity correlation function C(T) to detect the fractality of the crystal pattern [2]. Ihis equation is the expection value of the existence that two points separated by a distance r belong to the structure. In low concentration, Fig. 4 shows that the density-density correlation function within the clusters decays according to a power law. Furthermore, Fig. 4 shows that c(r) increases at higher concentration. 0.0
100
600
._ __ _
The success in understanding the transfer processes by external force in crystal growth has led to widespread interests from the technological point of view. An origin of curve of the branch in the metal leaf [8-lo] is a problem that should not be ignored. If the convection exists in the system, the branch of the metal leaf should be bent. If the pattern of the convection effect is investigated, these results [7] show that the branches have spiral form. The observed metal leaves are not spiral, and the curvature is uniform. Therefore, we propose the effect of Lorentz force in this crystal growth process. Sawada et al. [12] found several qualitatively different growth forms in electrochemical deposition of Zn from a thin layer of ZnS04 solution. The selfsimilarity of the crystal in electrochemical deposition disappears when applied voltage reaches a certain strength. Within the present model, we can reproduce that the pattern of the resultant crystal depends on the ion concentration and the applied voltage.
5. Conclusions -0.0
r
0.5
IlO
I
LA&
210
215 alo
Fig. 4. Double logarithmic plot of the density+Iensity tion function c(r) for multiparticle aggregation.
correla-
Using a Monte Carlo simulation, the pattern of metal leaf growths under high magnetic field and electrochemical deposition are investigated. The results are presented by comparing the patterns between the standard and perturbed DLA models arising from
50
H. Mizuseki
various environments. particle simulation this simulation,
The advantage
is the possibility
the effects of Coulomb
electrode
of this multithe ions. In
the number of the branches of crystal the Coulomb
affect the pattern
This information
force between the ions. forces between
ions and
and the self-similarity.
can not be obtained by single parti-
cle simulations.
Acknowledgements The authors would like to express sincere thanks to the crew of the Supercomputing
Center of Institute
for Materials Research, Tohoku University, for their continuous support of the HITAC S-3800/380 supercomputing
system.
Materials
Science
10 (1998) 4650
References
of investigating
forces between
is increased by the Coulomb Furthermore,
et al./Computational
[I] T.A. Witten and L.M. Sander, Phys. Rev. Lett. 47 (1981) 1400. [2] T. Vicsek, Fractal Growth Phenomena (World Scientitic, Singapore, 1992). [3] A. Sanchez, M.J. Bernal and J.M. Riveiro, Phys. Rev. E 50 (1994) R2427. [4] H. Mizuseki, K. Tanaka, K. Ohno and Y. Kawazoe, Sci. Rep. RITU A 43 (1997) 55. [5] R.F. Voss, Phys. Rev. B 30 (1984) 334. [6] P. Meakin, Phys. Rev. B 28 (1983) 5221. [7] L. Lopez-Tom& J. Claret, F. Mas and F. Sag&s, Phys. Rev. B 46 (1992) 11495. [S] I. Mogi, S. Okubo and Y. Nakagawa, J. Phys. Sot. Jpn. 60 (1991) 3200. [9] I. Mogi, S. Okubo and Y. Nakagawa, J. Cryst. Growth 128 (1993) 258. ]lO] 1. Mogi, M. Kamiko, S. Okubo and G. Kido, Physica B 201 (1994) 606. [ 111 M. Matsushita, M. Sano, Y. Hayakawa, H. Honjo and Y. Sawada, Phys. Rev. Lett. 53 (1984) 286. [ 121 Y. Sawada, A. Dougherty and J.P Gollub, Phys. Rev. L&t. 56 (1986) 1260.
ELSEVIER
Computational
Materials
Science
10 (1998) 4650
A new model of DLA under high magnetic field Hiroshi Mizuseki *, Kazumi Tanaka, Keiko Kikuchi, Kaoru Ohno, Yoshiyuki Kawazoe Institute jix Materials Research, Tohoku Universi@ Sendai 980-77, Japan
Abstract A new Monte Carlo model is introduced to describe diffusion-limited aggregation (DLA) with extra forces arising from Lorentz’s and/or Coulomb forces. Furthermore, we simulate a behavior of multiparticle diffusive aggregation to examine the resultant pattern of crystal in electrochemical deposition. Different patterns grown under various external forces are produced by Monte Carlo simulations. In the present model, the basic movement of particles is a random walk, with different transition probabilities in different directions, which characterizes stochastically the effect of extra forces. In case of assuming a high magnetic field, pattern formations which are qualitatively different from the standard DLA model are observed and they are successfully compared with preexisting experiments (Mogi et al., 1991). The present numerical results of electrochemical deposition show that the generated patterns strongly depend on the force acting on ions and their concentration (Sawada et al., 1986). Copyright 0 1998 Elsevier Science B.V. Keywords: Fractal dimension; deposition;
Diffusion-limited Monte Carlo method
aggregation
(DLA); Magnetic
1. Introduction The phenomena system
of crystal
have recently
growth
attracted
tion [ 1,2]. Pattern-formation
in diffusive
considerable
examples
atten-
in diffusive
system include electrochemical deposition, crystal growth, viscous fingering, dielectric breakdown, chemical dissolution and bacterial colonies. An approximation to these phenomena is provided by the Laplacian growth model that can be simulated by the diffusion-limited aggregation (DLA) [ 11. These patterns generated by DLA are analyzed by computational and experimental methods. Several extensions of the DLA model have been proposed to take into
* Corresponding author. Tel.: f81 22 215 2054; fax: +8 1 22 215 2052; e-mail: [email protected]. 0927-0256/98/$19.00
Copyright
PII SO927-0256(97)00142-O
0 1998 Elsevier
Science
field; Crystal
growth;
Random
walk; Electrochemical
account electrochemical deposition [3], magnetic field [4], concentration [5], drift [6], convection [7] and sticking probability [5]. A variety of computer simulations relationship
have been carried out to investigate the between the patterns of the crystal and
the movement behavior of the particles. Consequently, the morphology of the crystal strongly depends on the dynamics of the crystal growth process. As a new type of perturbation, Institute
applied a high magnetic
Mogi et al. at our field for the den-
dritic growth of silver [8], lead [9] and zinc [lo] in solutions
and obtained new specific patterns of metal
leaves. Under a high magnetic field, the branches bend slightly and the shape of the envelope becomes circular. These results imply that the particle diffusion in the medium is strongly affected by the magnetic field and this decisive difference in the particle movement
B.V. All rights reserved
H. Mizuseki et al. /Computational
changes the final pattern from the DLA result. Since the direction of bending is reversed, according to the reversal of magnetic field, the observed bending of the crystal is attributed to the cyclotron motion of the Ag ions. These experimental works have also shown that the effect of magnetic field is universal, i.e., independent of the details of the chemical properties of the solution. Therefore, we expect to reproduce this universality observed in the experiments via computer simulation. The first aim of this paper is to examine various patterns of crystal growths under an external force involving the stochastic treatment of the cyclotron motion. Sawada et al. [ 11,121 show that the patterns in electrochemical deposition depend on the concentration of the solution and applied voltage of electrode. This experiment shows that different pattern formations under various conditions such as an ion concentration and an applied voltage in the Zn from a thin layer of ZnSO4 solution appeared. It is worthwhile to examine the subject more closely. Therefore, we have to inquire, to some extent, into the force that is affected by other ions. In this paper, we present a new model for the electrodeposition process. The multiparticle DLA simulation is carried out to simulate this behavior of particle [ 11,121, because the particle is affected by Coulomb forces from the electrode as well as from the other ions. The present paper is organized as follows: A new growth model is introduced in the next section, which is affected by external force. A large computer simulation is performed and the obtained numerical results are given in Section 3. They are compared with the experimental observations also. Sections 4 and 5 are devoted, respectively, to discussion and conclusion.
2. Model and numerical method Our present model is basically a Monte Carlo simulation on a square lattice. The inclusion of external forces shows that the pattern formation is governed not only by the diffusion field, but also by the magnetic field, gradient of concentration, electric field, and so on. The effect of the magnetic field is included by
Materials Science 10 (1998) 46-50
47
simply changing the probability of the movement perpendicular to the Monte Carlo movement. We assume two environments in crystal growth: high magnetic field and gradient of ion concentration. Multiparticle aggregations have been simulated at various concentration f in the range of 0.05-0.4 and crystal sizes from 500 x 500 to 800 x 800. In the present simulation, when the particle arrives to contact with the crystal, it sticks permanently, and a sticking probability between the ion and the crystal is 1. Furthermore, we have not allowed that two ions occupy the same square sites. In this simulation, the area far from the electrode was held at initial concentration f. During simulations, f is fixed. This is essentially the model decribed previously [4] where it was used to study the effect of external force in crystal growth.
3. Results 3.1. High magnetic field Numerical simulations were performed based on the models described in the preceding section. lypical growth patterns obtained by the present research are indicated in Figs. 1 and 2 together with the experimental results [8]. The numbers on the axes show the mesh points and therefore the sizes of the model clusters. By the standard DLA growth model (Fig. l), the well-known pattern with the fractal dimension of 1.67 is obtained for the square lattice case. Fig. 2(b) shows the experimentally observed crystal pattern under high magnetic field [81. This figure bears resemblance to the present simulation from the view point of the bending of the pattern. The numerically obtained pattern in Fig. 2(a) is basically similar to the pattern of the experimental result. The numbers and the thickness of the branches in Figs. 2(a) and (b) are still different because of the small number of particles in simulation compared with experiment. If we can perform quantitatively larger scale simulations which are not possible at the moment, the number of branches increase and accordingly the branches will become thinner.
H. Mizuseki et al. /Computational Materials Science 10 (1998) 46-50
48
1000
1
I
I
500
0
-500
-y
30
I
I
I
-500
0
500
(a)
10011
fb)
Fig. 1. (a) Simulated pattern of the standard DLA model on square lattice. (b) Observed field. The square at the center of the metal leaf is a piece of copper metal [8].
pattern
of silver leaves without
magnetic
lOoF
l500
aI
-8
4
l-
I-
001D
0
(b) Fig. 2. (a) Simulated crystal growth with the effect of Lorentz force and gradient of electrolyte concentration, respectively. The conditions used are a = 1.O, 0 = IO’, and ionization tendency parameter b = 0.035. The notation of parameters was referred to [4]. (b) Observed pattern of silver leaves under 8T. The magnetic field was applied perpendicular to the plate [8].
H. Mizuseki et al. /Computational Materials Science 10 (1998) 46-50
49
600
100
200
400
300
500
600
100
200
300
400
500
Fig. 3. Multiparticle diffusive aggregation to point electrode from a concentration (a)
3.2. Electrochemical
I
,
I
i
I
-0.2-
-
k<
-0.4 -
--
f =
200
300
400
500
600
0.05, (b) f = 0.1 and (c) f = 0.2.
4. Discussion
deposition
Typical growth patterns obtained by the present research are indicated in Fig. 3. Figs. 3(a)-(c) show the results of aggregation from a point for different f. Naturally, in low concentration, the obtained pattern is very similar to the standard DLA pattern. On the other hand, in high concentration, the number of branches increases and accordingly the space between the branches becomes small. In the case of random structures such as present study, we need the so-called densitydensity correlation function C(T) to detect the fractality of the crystal pattern [2]. Ihis equation is the expection value of the existence that two points separated by a distance r belong to the structure. In low concentration, Fig. 4 shows that the density-density correlation function within the clusters decays according to a power law. Furthermore, Fig. 4 shows that c(r) increases at higher concentration. 0.0
100
600
._ __ _
The success in understanding the transfer processes by external force in crystal growth has led to widespread interests from the technological point of view. An origin of curve of the branch in the metal leaf [8-lo] is a problem that should not be ignored. If the convection exists in the system, the branch of the metal leaf should be bent. If the pattern of the convection effect is investigated, these results [7] show that the branches have spiral form. The observed metal leaves are not spiral, and the curvature is uniform. Therefore, we propose the effect of Lorentz force in this crystal growth process. Sawada et al. [12] found several qualitatively different growth forms in electrochemical deposition of Zn from a thin layer of ZnS04 solution. The selfsimilarity of the crystal in electrochemical deposition disappears when applied voltage reaches a certain strength. Within the present model, we can reproduce that the pattern of the resultant crystal depends on the ion concentration and the applied voltage.
5. Conclusions -0.0
r
0.5
IlO
I
LA&
210
215 alo
Fig. 4. Double logarithmic plot of the density+Iensity tion function c(r) for multiparticle aggregation.
correla-
Using a Monte Carlo simulation, the pattern of metal leaf growths under high magnetic field and electrochemical deposition are investigated. The results are presented by comparing the patterns between the standard and perturbed DLA models arising from
50
H. Mizuseki
various environments. particle simulation this simulation,
The advantage
is the possibility
the effects of Coulomb
electrode
of this multithe ions. In
the number of the branches of crystal the Coulomb
affect the pattern
This information
force between the ions. forces between
ions and
and the self-similarity.
can not be obtained by single parti-
cle simulations.
Acknowledgements The authors would like to express sincere thanks to the crew of the Supercomputing
Center of Institute
for Materials Research, Tohoku University, for their continuous support of the HITAC S-3800/380 supercomputing
system.
Materials
Science
10 (1998) 4650
References
of investigating
forces between
is increased by the Coulomb Furthermore,
et al./Computational
[I] T.A. Witten and L.M. Sander, Phys. Rev. Lett. 47 (1981) 1400. [2] T. Vicsek, Fractal Growth Phenomena (World Scientitic, Singapore, 1992). [3] A. Sanchez, M.J. Bernal and J.M. Riveiro, Phys. Rev. E 50 (1994) R2427. [4] H. Mizuseki, K. Tanaka, K. Ohno and Y. Kawazoe, Sci. Rep. RITU A 43 (1997) 55. [5] R.F. Voss, Phys. Rev. B 30 (1984) 334. [6] P. Meakin, Phys. Rev. B 28 (1983) 5221. [7] L. Lopez-Tom& J. Claret, F. Mas and F. Sag&s, Phys. Rev. B 46 (1992) 11495. [S] I. Mogi, S. Okubo and Y. Nakagawa, J. Phys. Sot. Jpn. 60 (1991) 3200. [9] I. Mogi, S. Okubo and Y. Nakagawa, J. Cryst. Growth 128 (1993) 258. ]lO] 1. Mogi, M. Kamiko, S. Okubo and G. Kido, Physica B 201 (1994) 606. [ 111 M. Matsushita, M. Sano, Y. Hayakawa, H. Honjo and Y. Sawada, Phys. Rev. Lett. 53 (1984) 286. [ 121 Y. Sawada, A. Dougherty and J.P Gollub, Phys. Rev. L&t. 56 (1986) 1260.