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Growth morphologies in diffusion fields
Journal of Crystal Growth 128 (1993) 82-86 North-Holland
j. . . . . . . .
CRYSTAL GROW
T H
Growth morphologies in diffusion fields Y u k i o Saito a, T o m o k o S a k i y a m a a and M a k i o U w a h a b,1 a Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223, Japan b Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980, Japan
Growth morphologies in diffusion fields are investigated by two types of numerical methods: solution of the integro-differential equation of a step profile and Monte Carlo simulation of a lattice gas model. The former method is applied to study the effect of kinetics on dendritic growth, and the capillarity-controlled scaling behavior is found to hold for a small but non-vanishing kinetic coefficient. The latter method is suitable for studying the effect of noise at the atomic level, and applied to the solidification by adsorption of atoms at the step. By an irreversible solidification, shot-noise is frozen in the aggregation to produce a fractal structure. Even when the noise is reduced, the step profile is found to show a spatio-temporal chaotic behavior after the straight step becomes unstable.
1. Introduction
The solidification front of a crystal shows a diversity of complex patterns, induced by the competition of various destabilizing and stabilizing factors [1]. The instability of the planar interface is caused by the diffusion field, and the linear stability analysis is done by Mullins and Sekerka [2]. In order to study the well-developed nonlinear structures one usually needs a small parameter to perform a perturbation analysis [1,3]. Under more general conditions, where small parameters are absent, numerical simulation might be the most appropriate method [4-14]. There are various simulation methods, among which we mention here two complementary ones: the solution of the integro-differential equation [6-9] and the lattice gas Monte Carlo simulation [10-14]. They are complementary in the sense that the former is derived from the continuous equation without noise, whereas the latter describes the discrete "atomic" system which contains noise at the atomic level.
] Present address: Department of Physics, College of General Education, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-01, Japan.
Noise is important in relation to the anisotropy of the system. It is by now well known that an anisotropy is essential in order to stabilize the tip of a dendritic crystal. In a viscous finger system, where an interface between two fluids of different viscosities moves in a Laplacian field, the interface shows an instability similar to that of a solidification system [4], but due to the absence of anisotropy an irregular shape results under the circular geometry of the Hele-Shaw cell [16]. Upon introducing anisotropy by imposing a groove on a glass plate covering the fluids, the fluid system shows dendritic structure [17,18]. On the contrary, by suppressing the anisotropy or enhancing the noise in a solidification system, the crystal shows a tip-splitting instability [19,20]. From these facts, the method of integro-differential equation is appropriate for the quantitative study of regular structures controlled by anisotropies, and the lattice gas model with noise has its advantage in the investigation of fluctuations or the noise-controlled irregular structures. The system considered here is a crystal growing in a diffusion field, u, which satisfies the diffusion equation
Ou / O t = DI7 2u.
0022-0248/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
(1)
Y. Saito et aL
/ Growth morphologies in diffusion fields
The diffusion field might be a thermal one for melt growth or a concentration one for solution growth. We here assume a one-sided model, where diffusion takes place only in the liquid phase. Far from the interface, the melt should be undercooled or the solution should be supersaturated to activate the solidification. The strength of the diffusion field u is measured from the value far from the interface. The energy or the mass conservation at the interface requires the relation between the interface growth velocity in the normal direction un and the normal flux of the diffusion field as Un = - D
On~On.
(2)
Also, for the interface to grow, an interracial undercooling or supersaturation is required, namely the diffusion field at the interface, ui, differs from the equilibrium value Ue q =
A
--
dK.
(3)
Here a is the undercooling, K is the curvature of the interface, and d is the capillary length related to the interface tension 7. For a rough interface, a linear response is expected between growth rate v, and interfacial undercooling, Ueq- ui, with a kinetic coefficient K as v n = g(Ueq - ui).
(4)
Generally the capillary length d and the kinetic coefficient K depend on the orientation of the interface, reflecting the crystalline anisotropy. We here assume the simple form with a four fold symmetry as d = do(1 - d 4 cos 40),
K -1 = b0(1 - b 4 COS 4 0 ) ,
83
tant fact found in these theoretical [1,3] and numerical [5,6] studies is the essential role of the anisotropy d 4 in the capillarity to stabilize the tip of the dendrite. In reality, however, the kinetic coefficient K is finite, and modification of the growth dynamics is expected [3,8].
2. Anisotropy controlled growth For a quantitative study of the kinetic effect on dendritic growth, we apply an integro-differential equation to the interface, since the strength of various parameters are easily controlled by this method [9]. The details of the study are described by Saito and Sakiyama [9], and the main result is summarized in fig. 1. For a finite range of small kinetic coefficient b0, the growth rate v or the stability parameter, o" = ( d o / 2 D p ( A ) 2 ) v , is independent of b 0 and solely determined by the capillarity d. Here the product of the growth rate v and the tip radius p, or the P6clet number p(A) = ( v p / 2 D ) is determined by the undercooling A through the Ivantsov relation A = ~
e p erfc(fp).
(6)
One also observes that cr is almost independent of A, representing the universal scaling relation
i
i
i
[
•
ee e
--3 • 0
O&
0
A
AO
AO
,k, O
(5)
A
b--4
v
A
AO
' O J 0.I 0.~5
for a two-dimensional crystal. For a rough interface, the atoms are smoothly incorporated in the solid, and the kinetic coefficient K is large. In the extreme case where K is infinite, local equilibrium at the interface, u i = Ueq, is realized. Many of the investigations have been done with this local equilibrium assumption [1,3], and a universal scaling relation is found between the growth rate and the structure of the dendrite. An impor-
i
• O
•
I
A JO.1 0.16 • •
--5
-6
I
0
1.0 0.~5 O
1.0 0.45
-
t4
-
I2
I 0
i 2
in Fig. 1. The stability parameter o - = ( d o / 2 D p ( A ) 2 ) v versus the normalized kinetics strength l ~ = 2 D p d ~ / 2 ( b o / d o) for various kinetic anisotropies v = ( b 4 / d 4 ) , and undercoo|ings A.
84
Y. Saito et al. / Growth morphologies in diffusion fields
Up 2 = constant. When the kinetic effect becomes
strong or b 0 becomes large, the growth rate v deviates from the value determined by the capillarity, and decreases in proportion to bo 1. Tip radius p becomes independent of the undercooling A or the growth velocity v, as is found in the viscous finger experiment [18]. The numerical solution of the integro-differential equation, however, cannot eliminate all noise. The system suffers from some uncontrolable numerical noise, caused probably by the discretization of the interface into many linear segments or the inaccuracy of the numerical integration. This numerical noise seems to be the cause of dendritic sidebranches [5,6], and also of the tip splitting, found for the cases of an isotropic capillarity [6], d 4 = 0, or of a very strong kinetic effect [9], b 0 >> 1, where the growth rate v becomes very small.
3. Noise controlled growth In order to investigate the effect of noise on the diffusional growth, a lattice gas Monte Carlo simulation is appropriate, since the physical meaning of noise is clear. The extreme case where the growth is controlled by the noise is realized at very low temperatures when the solidification occurs irreversibly [20]. Irreversible solidification means that solidified atoms will never melt again. This process freezes the shot-noise or the fluctuation associated with the diffusion process, and the resulting structure of the aggregate is irregular [21]. In our previous simulation [11] of the aggregation growth from a finite density lattice gas, the aggregate is found to have a fractal structure similar to the diffusion-limited aggregation (DLA) [21] for a short length scale, r < l, whereas it takes a homogeneous, compact structure for a large scale, r > l. The growth rate v is proportional to some power of the gas density ng at low densities. The crossover from the DLA-like structure to the compact one is caused by the global mass conservation. The growth dynamics is explained by assuming that the structural crossover takes place at a diffusion length l = 2 D / v , which defines the range of gas density suppression
around the aggregate. One obtains the power law relation between v and ng as v = an 1~(a-DO,
(7)
with a constant a, d the spatial dimension, and D e the fractal dimension of the DLA. The value of Df obtained by the simulation agrees with that obtained by a large scale simulation study of the pure D L A [22]. If the diffusion length l really controls the growth, an external perturbation of the diffusion length will modify the dynamics of the system. For that purpose we add a drift flow parallel to the growth direction. Above a critical drift velocity the steady growth of an aggregation is found to be impossible [12]. The transition from a steady growth to a non-growth state is first-order. The transverse flow does not affect the vertical growth rate but it tilts the whole aggregate towards the direction of the drift flow [13]. The reduction of noise or the enhancement of anisotropy is expected to produce more regular patterns. This can be accomplished by allowing melting at the solidification front, and a crossover from an irregular dendrite to a regular one is found [10]. For a very slow growth, even a faceted polyhedral crystal is obtained.
4. Instability of a step morphology The irregular dendritic structure has been observed in molecular b e a m epitaxy (MBE) on a cold substrate [20]. In the M B E experiment or, more generally, in crystal growth from the vapor, diffusing gas atoms do not stay in the adsorbed layer above the substrate, but they can evaporate into or impinge from the ambient vapor. The impingement rate f and the life time ~ of the adsorbed atom modify the diffusion equation as OU/Ot = D~72u - u//'r,
(8)
where u(r, t ) = c ( r , t ) - 7 f . The last term u / ~ introduces a new length scale, which is also called the diffusion length x s -DvrD~, but is independent of the growth rate v. The system is characterized by the smaller of the two, x s and l =
Y. Saito et al. / Growth morphologies in diffusion fields
..:.', .;.,,...',..' ::...: ~.', :.;" .~...;::.::..?:.:."...~,,~.:~:,,..::.::,:..~:',~...:.-.~:;..:: "" .: :: ..'.,:?."" .. :' ,. "r :.~.'i: ;:. ~,',?. ".. : ":'~::~,~":.,".~ : "' k;i.':.-. .~.'. :.:?"-.... \ .':'. '.. :.'.....~,:,...', ~..: ;....,~.~'. :'~:: .." .. :~L.~q...' ~.~.,"
".. : ." v.,''. ""',C'.' ":":..' "':~;"
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., .
.
.
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.
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•
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.;:.:.:..
.
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'
,~
85
The step stiffness stabilizes the straight step when the growth is slow, whereas the straight step becomes unstable for a fast growth at f > f c , where fc is the critical impingement frequency for the diffusional Mullins-Sekerka instability [23]. The details of the model and the behavior of the fluctuation for f < fc are described by Uwaha and Saito [14]. H e r e we concentrate on the pattern formation after the instability has occurred. The linear stability analysis of the straight step with a sinusoidal modification t~y(x, t) ~ e °~(k)t sin kx
(9)
leads to the dispersion relation t o ( k ) = vo( f - f ~ ) k 2 - tzk 4.
Fig. 2. Irreversibly solidified step profile at T = 0 with a finite i m p i n g i n g rate f a n d the e v a p o r a t i o n r a t e z. System size is 256 x 256, and the diffusion l e n g t h is Xs -= D ~ 7 = 32.
2 D / v . Our previous simulation corresponds to the case of f = 0 and r = oo with a constant coo f r , or x s = oo. For a finite x s, the irreversible solidification at a step at the zero t e m p e r a t u r e produces an aggregate, as shown in fig. 2. At the first glance, the structure seems to be similar to that obtained previously without impingement and detachment of adsorbed atoms. But the fractal structure is realized only in the front region of the solidification, and its width is of the order of xs. The aggregation behind this region thickens due to the perpetual impingement of the gas atoms from vapor. At finite temperatures, a solid atom at the step can sublimate from the step onto the lower terrace and behaves as a diffusing adsorbed atom [14,23]. The rate of sublimation or solidification at the step is determined by the kink energy and the chemical potential difference of the crystal and the adsorbed state. The multiple sublimation and the solidification processes at a step site average out the shot-noise of the atomic level. The step profile becomes smooth, and reflects the step stiffness determined by the kink energy.
(10)
The amplification rate to(k) becomes positive for a band of wavevectors k at f > fc. One expects that for f > f c , the nonlinear structure may be periodic with the wave vector kmax, which corresponds to the maximum of to(k). From the Monte Carlo simulation, however, the step profile is not
Fig. 3. Stroboscopic t i m e e v o l u t i o n of an u n s t a b l e step, which shows a s p a t i o - t e m p o r a l chaotic behavior. System size is 256 x 2 5 6 , e n e r g y p a r a m e t e r s are E / T = ( a / T = 2.0, and the diffusion l e n g t h is x s --- Dx/-b-~-r= 16.
86
Y.. Saito et al. / Growth morphologies in diffusion fields
stationary, as is shown in the stroboscopic picture of the time evolution of the step front (fig. 3). The step has deep valleys, whose separation is of the order of the periodicity 2~r/kma x of the most unstable mode, but the structure is quite far from regular. It is distinctly different from the regular and periodic structure found in the cellular structure of directional solidification. Valleys also move laterally, and the direction of the motion is not definite to be right or left. If the two valleys move in different directions, they may collide and disappear, and the hill between the separating two valleys widens and the flat top splits to create a new valley. This spatio-temporal chaotic behavior seems to be induced by the noise still remaining in the process. But near the critical point fc, the present model can be cast [24] into the Kuramoto-Sivashinsky (KS) equation [25], which is a deterministic equation without noise and still is known to have the spatio-temporal chaotic behavior [26]. Then the erratic behavior found might not be due to the effect of the noise but the intrinsic phenomenon. This example shows that the reduction of the noise does not neccessarily lead to a regular behavior. Since the KS equation is valid, in principle, only near the critical point, one should use an integro-differential method for the general situation far from the critical point fc in order to prove that the apparent chaotic behavior of the step profile is not caused by the noise.
Acknowledgements Part of this work is supported by a Grant-inAid for Scientific Research on Priority Areas of the Ministry of Science, Education and Culture with the contract No. 03243101. Y.S. acknowledges C. Misbah for enlightening discussions. He also acknowledges the hospitality of the Institut Laue-Langevin in Grenoble and the Groupe de Physique des Solides, Universit6 Paris 7, where part of this work was done.
References [1] J. S. Langer, in: Chance and Matter, Eds. J. Souletie, J. Vannimenus and R. Stora (North-Holland, Amsterdam, 1987). [2] W.W. Mullins and R.F. Sekerka, J. Appl. Phys. 35 (1964) 444. [3] E.A. Brener and V.I. Mel'nikov, Advan. Phys. 40 (1991) 53. [4] L.H. Ungar and R.A. Brown, Phys. Rev. B 29 (1984) 1367. [5] D. Kessler, J. Koplik and H. Levine, Advan. Phys. 37 (1988) 255. [6] Y. Saito, G. Goldbeck-Wood and H. Miiller-Krumbhaar, Phys. Rev. Letters 58 (1987) 1541; Phys. Rev. A 38 (1988) 2148. [7] C. Misbah, H. Miiller-Krumbhaar and Y. Saito, J. Crystal Growth 99 (1990) 156. [8] A. Classen, C. Misbah, H. Miiller-Krumbhaar and Y. Saito, Phys. Rev. A 43 (1991) 6920. [9] Y. Saito and T. Sakiyama, J. Crystal Growth 128 (1993) 224. [10] Y. Saito and T. Ueta, Phys. Rev. A 40 (1989) 3408. [11] M. Uwaha and Y. Saito, J. Phys. Soc. Japan 51 (1988) 3285; Phys. Rev. A 40 (1989) 4716. [12] S. Seki, M. Uwaha and Y. Saito, Europhys. Letters 14 (1991) 397. [13] Y. Saito, M. Uwaha and S. Seki, in: Dynamical Phenomena at Interfaces, Eds. S.H. Davis and Huppert (Kluwer, Dordrecht, 1992). [14] M. Uwaha and Y. Saito, Phys. Rev. Letters 68 (1992) 224; J. Crystal Growth 128 (1993) 87. [15] P.G. Saffman and G.I. Taylor, Proc. Roy. Soc. (London) A 245 (1958) 312. [16] L. Paterson, J. Fluid Mech. 113 (1981) 513. [17] E. Ben-Jacob, R. Godbey, N.D. Goldenfeld, J. Koplik, H. Levine, T. Mueller and L.M. Sander, Phys. Rev. Letters 55 (1985) 1315. [18] M. Matsushita and H. Yamada, J. Crystal Growth 99 (1990) 161. [19] H. Honjo, S. Ohta and M. Matsushita, J. Phys. Soc. Japan 55 (1986) 2487. [20] R.Q. Hwang, J. Schr6der, C. Giinter and R.J. Behm, Phys. Rev. Letters 67 (1991) 3279. [21] T.A. Witten and L.M. Sander, Phys. Rev. Letters 47 (1981) 1400; Phys. Rev. B 27 (1983) 5686. [22] S. Tolman and P. Meakin, Phys. Rev. A 40 (1989) 428. [23] G.S. Bales and A. Zangwill, Phys. Rev. B 41 (1990) 5500. [24] I. Bena, C. Misbah and A. Valance, unpublished. [25] Y. Kuramoto and T. Tsuzuki, Progr. Theoret. Phys. 55 (1976) 356; G.I. Sivashinsky, Acta Astronautica 4 (1977) 1177. [26] P. Manneville, Dissipative Structures and Weak Turbulence (Academic Press, New York, 1990).
j. . . . . . . .
CRYSTAL GROW
T H
Growth morphologies in diffusion fields Y u k i o Saito a, T o m o k o S a k i y a m a a and M a k i o U w a h a b,1 a Department of Physics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223, Japan b Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980, Japan
Growth morphologies in diffusion fields are investigated by two types of numerical methods: solution of the integro-differential equation of a step profile and Monte Carlo simulation of a lattice gas model. The former method is applied to study the effect of kinetics on dendritic growth, and the capillarity-controlled scaling behavior is found to hold for a small but non-vanishing kinetic coefficient. The latter method is suitable for studying the effect of noise at the atomic level, and applied to the solidification by adsorption of atoms at the step. By an irreversible solidification, shot-noise is frozen in the aggregation to produce a fractal structure. Even when the noise is reduced, the step profile is found to show a spatio-temporal chaotic behavior after the straight step becomes unstable.
1. Introduction
The solidification front of a crystal shows a diversity of complex patterns, induced by the competition of various destabilizing and stabilizing factors [1]. The instability of the planar interface is caused by the diffusion field, and the linear stability analysis is done by Mullins and Sekerka [2]. In order to study the well-developed nonlinear structures one usually needs a small parameter to perform a perturbation analysis [1,3]. Under more general conditions, where small parameters are absent, numerical simulation might be the most appropriate method [4-14]. There are various simulation methods, among which we mention here two complementary ones: the solution of the integro-differential equation [6-9] and the lattice gas Monte Carlo simulation [10-14]. They are complementary in the sense that the former is derived from the continuous equation without noise, whereas the latter describes the discrete "atomic" system which contains noise at the atomic level.
] Present address: Department of Physics, College of General Education, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-01, Japan.
Noise is important in relation to the anisotropy of the system. It is by now well known that an anisotropy is essential in order to stabilize the tip of a dendritic crystal. In a viscous finger system, where an interface between two fluids of different viscosities moves in a Laplacian field, the interface shows an instability similar to that of a solidification system [4], but due to the absence of anisotropy an irregular shape results under the circular geometry of the Hele-Shaw cell [16]. Upon introducing anisotropy by imposing a groove on a glass plate covering the fluids, the fluid system shows dendritic structure [17,18]. On the contrary, by suppressing the anisotropy or enhancing the noise in a solidification system, the crystal shows a tip-splitting instability [19,20]. From these facts, the method of integro-differential equation is appropriate for the quantitative study of regular structures controlled by anisotropies, and the lattice gas model with noise has its advantage in the investigation of fluctuations or the noise-controlled irregular structures. The system considered here is a crystal growing in a diffusion field, u, which satisfies the diffusion equation
Ou / O t = DI7 2u.
0022-0248/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
(1)
Y. Saito et aL
/ Growth morphologies in diffusion fields
The diffusion field might be a thermal one for melt growth or a concentration one for solution growth. We here assume a one-sided model, where diffusion takes place only in the liquid phase. Far from the interface, the melt should be undercooled or the solution should be supersaturated to activate the solidification. The strength of the diffusion field u is measured from the value far from the interface. The energy or the mass conservation at the interface requires the relation between the interface growth velocity in the normal direction un and the normal flux of the diffusion field as Un = - D
On~On.
(2)
Also, for the interface to grow, an interracial undercooling or supersaturation is required, namely the diffusion field at the interface, ui, differs from the equilibrium value Ue q =
A
--
dK.
(3)
Here a is the undercooling, K is the curvature of the interface, and d is the capillary length related to the interface tension 7. For a rough interface, a linear response is expected between growth rate v, and interfacial undercooling, Ueq- ui, with a kinetic coefficient K as v n = g(Ueq - ui).
(4)
Generally the capillary length d and the kinetic coefficient K depend on the orientation of the interface, reflecting the crystalline anisotropy. We here assume the simple form with a four fold symmetry as d = do(1 - d 4 cos 40),
K -1 = b0(1 - b 4 COS 4 0 ) ,
83
tant fact found in these theoretical [1,3] and numerical [5,6] studies is the essential role of the anisotropy d 4 in the capillarity to stabilize the tip of the dendrite. In reality, however, the kinetic coefficient K is finite, and modification of the growth dynamics is expected [3,8].
2. Anisotropy controlled growth For a quantitative study of the kinetic effect on dendritic growth, we apply an integro-differential equation to the interface, since the strength of various parameters are easily controlled by this method [9]. The details of the study are described by Saito and Sakiyama [9], and the main result is summarized in fig. 1. For a finite range of small kinetic coefficient b0, the growth rate v or the stability parameter, o" = ( d o / 2 D p ( A ) 2 ) v , is independent of b 0 and solely determined by the capillarity d. Here the product of the growth rate v and the tip radius p, or the P6clet number p(A) = ( v p / 2 D ) is determined by the undercooling A through the Ivantsov relation A = ~
e p erfc(fp).
(6)
One also observes that cr is almost independent of A, representing the universal scaling relation
i
i
i
[
•
ee e
--3 • 0
O&
0
A
AO
AO
,k, O
(5)
A
b--4
v
A
AO
' O J 0.I 0.~5
for a two-dimensional crystal. For a rough interface, the atoms are smoothly incorporated in the solid, and the kinetic coefficient K is large. In the extreme case where K is infinite, local equilibrium at the interface, u i = Ueq, is realized. Many of the investigations have been done with this local equilibrium assumption [1,3], and a universal scaling relation is found between the growth rate and the structure of the dendrite. An impor-
i
• O
•
I
A JO.1 0.16 • •
--5
-6
I
0
1.0 0.~5 O
1.0 0.45
-
t4
-
I2
I 0
i 2
in Fig. 1. The stability parameter o - = ( d o / 2 D p ( A ) 2 ) v versus the normalized kinetics strength l ~ = 2 D p d ~ / 2 ( b o / d o) for various kinetic anisotropies v = ( b 4 / d 4 ) , and undercoo|ings A.
84
Y. Saito et al. / Growth morphologies in diffusion fields
Up 2 = constant. When the kinetic effect becomes
strong or b 0 becomes large, the growth rate v deviates from the value determined by the capillarity, and decreases in proportion to bo 1. Tip radius p becomes independent of the undercooling A or the growth velocity v, as is found in the viscous finger experiment [18]. The numerical solution of the integro-differential equation, however, cannot eliminate all noise. The system suffers from some uncontrolable numerical noise, caused probably by the discretization of the interface into many linear segments or the inaccuracy of the numerical integration. This numerical noise seems to be the cause of dendritic sidebranches [5,6], and also of the tip splitting, found for the cases of an isotropic capillarity [6], d 4 = 0, or of a very strong kinetic effect [9], b 0 >> 1, where the growth rate v becomes very small.
3. Noise controlled growth In order to investigate the effect of noise on the diffusional growth, a lattice gas Monte Carlo simulation is appropriate, since the physical meaning of noise is clear. The extreme case where the growth is controlled by the noise is realized at very low temperatures when the solidification occurs irreversibly [20]. Irreversible solidification means that solidified atoms will never melt again. This process freezes the shot-noise or the fluctuation associated with the diffusion process, and the resulting structure of the aggregate is irregular [21]. In our previous simulation [11] of the aggregation growth from a finite density lattice gas, the aggregate is found to have a fractal structure similar to the diffusion-limited aggregation (DLA) [21] for a short length scale, r < l, whereas it takes a homogeneous, compact structure for a large scale, r > l. The growth rate v is proportional to some power of the gas density ng at low densities. The crossover from the DLA-like structure to the compact one is caused by the global mass conservation. The growth dynamics is explained by assuming that the structural crossover takes place at a diffusion length l = 2 D / v , which defines the range of gas density suppression
around the aggregate. One obtains the power law relation between v and ng as v = an 1~(a-DO,
(7)
with a constant a, d the spatial dimension, and D e the fractal dimension of the DLA. The value of Df obtained by the simulation agrees with that obtained by a large scale simulation study of the pure D L A [22]. If the diffusion length l really controls the growth, an external perturbation of the diffusion length will modify the dynamics of the system. For that purpose we add a drift flow parallel to the growth direction. Above a critical drift velocity the steady growth of an aggregation is found to be impossible [12]. The transition from a steady growth to a non-growth state is first-order. The transverse flow does not affect the vertical growth rate but it tilts the whole aggregate towards the direction of the drift flow [13]. The reduction of noise or the enhancement of anisotropy is expected to produce more regular patterns. This can be accomplished by allowing melting at the solidification front, and a crossover from an irregular dendrite to a regular one is found [10]. For a very slow growth, even a faceted polyhedral crystal is obtained.
4. Instability of a step morphology The irregular dendritic structure has been observed in molecular b e a m epitaxy (MBE) on a cold substrate [20]. In the M B E experiment or, more generally, in crystal growth from the vapor, diffusing gas atoms do not stay in the adsorbed layer above the substrate, but they can evaporate into or impinge from the ambient vapor. The impingement rate f and the life time ~ of the adsorbed atom modify the diffusion equation as OU/Ot = D~72u - u//'r,
(8)
where u(r, t ) = c ( r , t ) - 7 f . The last term u / ~ introduces a new length scale, which is also called the diffusion length x s -DvrD~, but is independent of the growth rate v. The system is characterized by the smaller of the two, x s and l =
Y. Saito et al. / Growth morphologies in diffusion fields
..:.', .;.,,...',..' ::...: ~.', :.;" .~...;::.::..?:.:."...~,,~.:~:,,..::.::,:..~:',~...:.-.~:;..:: "" .: :: ..'.,:?."" .. :' ,. "r :.~.'i: ;:. ~,',?. ".. : ":'~::~,~":.,".~ : "' k;i.':.-. .~.'. :.:?"-.... \ .':'. '.. :.'.....~,:,...', ~..: ;....,~.~'. :'~:: .." .. :~L.~q...' ~.~.,"
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85
The step stiffness stabilizes the straight step when the growth is slow, whereas the straight step becomes unstable for a fast growth at f > f c , where fc is the critical impingement frequency for the diffusional Mullins-Sekerka instability [23]. The details of the model and the behavior of the fluctuation for f < fc are described by Uwaha and Saito [14]. H e r e we concentrate on the pattern formation after the instability has occurred. The linear stability analysis of the straight step with a sinusoidal modification t~y(x, t) ~ e °~(k)t sin kx
(9)
leads to the dispersion relation t o ( k ) = vo( f - f ~ ) k 2 - tzk 4.
Fig. 2. Irreversibly solidified step profile at T = 0 with a finite i m p i n g i n g rate f a n d the e v a p o r a t i o n r a t e z. System size is 256 x 256, and the diffusion l e n g t h is Xs -= D ~ 7 = 32.
2 D / v . Our previous simulation corresponds to the case of f = 0 and r = oo with a constant coo f r , or x s = oo. For a finite x s, the irreversible solidification at a step at the zero t e m p e r a t u r e produces an aggregate, as shown in fig. 2. At the first glance, the structure seems to be similar to that obtained previously without impingement and detachment of adsorbed atoms. But the fractal structure is realized only in the front region of the solidification, and its width is of the order of xs. The aggregation behind this region thickens due to the perpetual impingement of the gas atoms from vapor. At finite temperatures, a solid atom at the step can sublimate from the step onto the lower terrace and behaves as a diffusing adsorbed atom [14,23]. The rate of sublimation or solidification at the step is determined by the kink energy and the chemical potential difference of the crystal and the adsorbed state. The multiple sublimation and the solidification processes at a step site average out the shot-noise of the atomic level. The step profile becomes smooth, and reflects the step stiffness determined by the kink energy.
(10)
The amplification rate to(k) becomes positive for a band of wavevectors k at f > fc. One expects that for f > f c , the nonlinear structure may be periodic with the wave vector kmax, which corresponds to the maximum of to(k). From the Monte Carlo simulation, however, the step profile is not
Fig. 3. Stroboscopic t i m e e v o l u t i o n of an u n s t a b l e step, which shows a s p a t i o - t e m p o r a l chaotic behavior. System size is 256 x 2 5 6 , e n e r g y p a r a m e t e r s are E / T = ( a / T = 2.0, and the diffusion l e n g t h is x s --- Dx/-b-~-r= 16.
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stationary, as is shown in the stroboscopic picture of the time evolution of the step front (fig. 3). The step has deep valleys, whose separation is of the order of the periodicity 2~r/kma x of the most unstable mode, but the structure is quite far from regular. It is distinctly different from the regular and periodic structure found in the cellular structure of directional solidification. Valleys also move laterally, and the direction of the motion is not definite to be right or left. If the two valleys move in different directions, they may collide and disappear, and the hill between the separating two valleys widens and the flat top splits to create a new valley. This spatio-temporal chaotic behavior seems to be induced by the noise still remaining in the process. But near the critical point fc, the present model can be cast [24] into the Kuramoto-Sivashinsky (KS) equation [25], which is a deterministic equation without noise and still is known to have the spatio-temporal chaotic behavior [26]. Then the erratic behavior found might not be due to the effect of the noise but the intrinsic phenomenon. This example shows that the reduction of the noise does not neccessarily lead to a regular behavior. Since the KS equation is valid, in principle, only near the critical point, one should use an integro-differential method for the general situation far from the critical point fc in order to prove that the apparent chaotic behavior of the step profile is not caused by the noise.
Acknowledgements Part of this work is supported by a Grant-inAid for Scientific Research on Priority Areas of the Ministry of Science, Education and Culture with the contract No. 03243101. Y.S. acknowledges C. Misbah for enlightening discussions. He also acknowledges the hospitality of the Institut Laue-Langevin in Grenoble and the Groupe de Physique des Solides, Universit6 Paris 7, where part of this work was done.
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