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Wavelength selection of cellular patterns
Scripta
METALLURGICA
Vol. 22, pp. 8 9 3 - 8 9 8 , 1988 P r i n t e d in the U . S . A .
Pergamon
Press
plc
WAVELENGTH SELECTION OF CELLULAR PATTERNS M. A. Eshelman and R. Trivedi Ames Laboratory-USDOE and the Department of Materials Science and Engineering Iowa State University, Ames, IA 50011
(Received March 7, 1988)
Introduction The phenomenon of pattern formation in systems which are driven far away from thermodynamic equilibrium has received an increasing amount of theoretical interest in a variety of scientific disciplines [I-2]. One of the important systems, which we shall consider in this paper, is the directional solidification of a two-component system. In directional solidification, an alloy of fixed composition is solidified at a desired velocity under the condition of a constant positive temperature gradient in the liquid. At low velocities a planar solid-liquid interface exists. As the velocity is increased above some critical value, Vc, the planar interface becomes unstable and reorganizes into a periodic array of cells. When a planar interface is driven just beyond the critical velocity, Vc, experimental studies show that the the steady-state cellular structure which emerges has a definite wavelength and amplitude whose magnitudes depend on the value of the steady-state velocity [3-8]. Theoretical models, based on the solvability condition, however, predict a discrete set of wavelengths [9]. The principle which selects a specific wavelength from a set of possible wavelengths is not yet established, but it is generally believed that the selection process occurs in a highly nonlinear regime which allows the elimination or creation of cells to achieve the final spacing [8,10]. In order to obtain a better understanding of the dynamics of the wavelength selection process, we have carried out directional solidification studies in the succinonitrile - 0.15 wt.% acetone system. This system is transparent so that the dynamics of the pattern reorganization can be studied in situ. The changes in the interface shape with time were observed and analyzed by Fourier transforms. From the Fourier spectra, the dynamical changes in the pattern wavenambers with time were evaluated to determine the process by which the pattern selection occurs. Experimental Directional solidification studies were carried out in an apparatus which is described by Somboonsuk et al. [8] and Mason and Eshelman [Ii]. Both succinonitrile and acetone were first purified and then mixed under an inert environment to obtain a mixture with 0.15 wt.% acetone. The temperature gradient stage was thermally equilibrated to give a temperature gradient of 3.76 K/-,, at the interface. The system was directionally solidified at a velocity of 0.5 ~m/s for 210 min. During this time a steady-state planar interface was established. The velocity was then increased to 0.7 ~m/s for 18 min. and the interface was found to remain planar. The velocity was increased further to 0.8 ~m/s, where the planar interface became unstable after II min. The shape of the interface was photographed at 30 second intervals for 120 minutes. From the photomicrographs, the interface shape was digitized, and the resultant data set was input into a VAX computer for analysis. Figure 1 shows the changes in the interface shape as a function of time for a few selected times. The time at which the first distinguishable sign of instability occurred was taken to be t = O. The evolution of the interface pattern was analyzed in the wave vector space by Fourier transforming the digitized interface shapes for each time step. Within a small time range, some of the cellular perturbations folded back upon itself. The overhang of such cells was neglected, but the average location and the amplitude of these cells were not changed. In order to reduce
893 0036-9748/88 $3.00
+ .00
894
CELLULAR
3.5
i
i
i
i
PATTERNS
i
i
i
i
i
Vol.
i
22,
No.
6
i
5.0 ~
2.5 2.0 E E1
.
5
LO 0.5 0.5
1.0
1.5
2.0 2.5
mm
FIG. i.
Time evolution of an interface pattern in a stationary frame of reference. The interface profiles shown here are digitalized profiles of the experimental interface shapes observed in the directionally solidified succinonitrile - 0.15 at% acetone system at G ffi 3.76 k/mm and V = 0.8 ~m/s.
noise in the measurements, the input data set was smoothed with a three point center weighted smoothing function. The digitized data set, after the smoothing, was found to reproduce the interface shape quite accurately. Specific details of the Fourier analysis of interface shapes is given by Eshelman [12]. Experiments were also repeated to check the reproducibility of the results. Results Spatial Fourier transforms of patterns at times shortly after the occurrence of instability are shown in Fig. 2. The initial instability occurs with a peak at the waven~nber of 0.0708 ~m -I. There is also a peak at the wavent~aber of 0.003 ~m -I, but it is diffused due to the bias at the origin. This peak corresponds to the long wavelength perturbations which can be seen in Fig. I. The existence of two characteristic wavenumbers for the initial instability of the interface was also discussed by Trivedi and Somboonsuk [13]. Since the low wave number peak disappears in the cellular growth region once the interface is uniformly perturbed, we will not discuss it further. The general characteristics of the spectrum do not change significantly between 0 and 2 minutes, except that the peak at k = 0.0708 ~m -I shifts to the left, This shift corresponds to the slight increase in spacing which occurs between the pair of perturbations shown in Fig. I. In addition to the sh~ft, small peaks begin to appear at low wavenwaber values. A distinct change in the spectrum occurs between 2 and 3 minutes. We shall show later on that in this time range nonlinear effects become important. Figure 2b shows the Fourier spectra for t ffi 0, 5, i0, 15 and 20 minutes after the initial instability. Two important observations can be made: (I) a finite number of peaks is observed, and (2) the peak which corresponds to the lowest wavenumber begins to amplify at the largest rate. Figure 2c shows the spectra after 50 minutes. Note that one peak becomes prominent, indicating that the pattern is approaching the steady-state wavelength.
Vol.
22,
No.
6
CELLULAR
.10
PATTERNS
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896
CELLULAR
PATTERNS
Vol.
22,
No.
6
Discussion The spectral analysis of the interface shape gives insight into the dynamical nature of the pattern formation. The initial peak, which forms in the linear regime, amplifies more slowly compared to the subsequent peaks which form in the nonlinear regime. In order to examine the amplification rate of these various wavenumber peaks, the amplitude of the Fourier spectrum is plotted as a function of time for three characteristic peaks, as shown in Fig. 3. Three wavenumbers were chosen which represent the initial, intermediate and final wavenumber of the pattern. The initial peak, which corresponds to k = 0.0708 pm -I, shows a linear behavior up to about 2 minutes. After 2 minutes a sharp nonlinear behavior is observed. The amplitude of this peak begins to grow slowly as other peaks develop. The intermediate peak (k = 0.0507 Bm -I) amplifies at a very rapid rate and competes with the final peak until about 20 minutes. The final peak, which corresponds to k = 0.0199 pm -I becomes predominant only after 20 minutes. The final peak has a significant width (Fig. 2c) which shows that the wavelength selection criterion is not very sharp. The general characteristics of the wave number evolutions was found to be reproducible under different experimental conditions in this system as well as in the pivalic acid - ethanol system [14]. The pattern selection process can be divided into linear and nonlinear regions. In the linear region, one wavenumber predominates. Development of new peaks at lower wavenumber occurs when nonlinear effect become important. In the nonlinear regime, several peaks exist over a period of time. Since the differences in amplitudes of these peaks are small, it is not quite certain whether the middle three peaks are real or not. They could be represented by one wider peak. Several similar experiments, however, gave rise to similar peaks so that the existence of several intermediate peaks is indeed experimentally reproducible. At longer times, Fig. 2C, the lowest wavenumber peak begins to amplify sharply. This rapid amplification occurs because highly nonlinear effects cause cell elimination which allows the pattern to decrease the wavenumber
[8]. The wavenumber of the initial unstable pattern was found to be 0.07070 pm -I, which is somewhat larger the value of 0.017 pm -I predicted by the linear stability analysis of Mullins and Sekerta [15]. Since the amplication rate of the initial waven~nber is found to show a linear behaviour, fig. 3, one would expect the linear theory to predict the experimental observation. However, linear theory is based on the interface instability at a constant velocity. In our experiments, the initial instability was observed when the external velocity was changed from 0.7
1.0
-o- k:0.0199 ' --a-- k:O.0507 ---0.- k:0.0707
' .~
' ~
'/~'~'~t / " / [
=- 0.I o. E
0.01 O0
FIG. 3.
5.0
I0.0
15.0 Time,rain.
20.0
2.5.0
The variation in amplitude with time for selected wavenumbers. The wavenumber, k = 0.0199 pm -I, which has a smaller initial amplitude, becomes predominant at larger times when nonlinear effects become important.
Vol.
22,
No.
6
CELLULAR
PATTERNS
897
to 0.8 ~m/s. Since the experimental system is transparent, it was possible to measure the change in interface velocity with time and to identify the interface velocity at which the instability occurred. Figure 4 shows the variation in the interface velocity with time. In order to compare our results with the theory, it is important to note that the linear stability analysis is based on the steady-state solute profile in liquid. Since the velocity changes sharply in our experiments, dynamical effects become important which cause the concentration gradient at the interface to be larger than that predicted by the steady-state analysis. Eshelman and Trivedi [16] have examined this effect experimentally and shown that the dynamical results can be approximately compared with the steady-state model by making appropriate corrections in the velocity or composition value. These corrected velocity-wave nomber results are shown in Fig. 5 and compared with the theoretical prediction of Mullins and Sckerka [15]. The initial wavenumber could be measured only for the profile that was observed a few seconds after the break-up, at which time the velocity of the interface was slightly above Ve, as measured from Fig. 4. This result is shown as an open circle in Fig. 5, and it is close to the fastest growing wavenumber at that velocity. This experimental difficulty in observing the wavent~ber at the critical velocity may be responsible for experimental results which show initial w a v e n ~ b e r to be larger than that predicted by the linear stability analysis [7, 16]. Since the curvature of the V-k plot is very small at the critical velocity, a slight increase in velocity will significantly alter the wavenumber. As the velocity of the interface increases sharply with time, nonlinear effects become important which give rise to w a v e n ~ b e r peaks shown by crosses in Fig. 5. The final spacing, established after 50 minutes is shown as a filled circle. This spacing was found to continue to exist even when steady-state conditions were established. According to the linear theory, the planar interface should be stable at 0.8~m/s under steady-state conditions. However~ as shown by Eshelman and Trivedi [15], the planar to cellular transition in the succinonitrite-acetone system is subcritical so not the spacings established at V > V c will continue to be stable at V slightly below V c. These experiment results show that, as nonlinear effects become important, k decreases so that the time evolution of cellular pattern is accompanied by the coarsening of the initial pattern.
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lanor
0.5
FIG. 4.
I I
I
0
Nonplanor
I
I0
I
I
20 50
I
i
I
40 50 60 Time, min
70 80
The variations in interface velocity with time when the external velocity was changed from 0.7 to 0.8 ~m/s.
898
CELLULAR
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Vol.
22,
No.
6
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/
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_~ 2.o
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PATTERNS
o tnitiob wave n u m b e r
I.O
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LO-3
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I I0 "1
I I
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Wave number, /~m" I
FIG. 5.
Relationship between velocity and waven,~ber, as predicted by the linear stability analysis, for succinonitrile - 0.15 wt% acetone alloys at G = 3.76 K/mm. The line between the experimental data prints show the velocity range over which the given wavenumber was predominant.
Conclusions The process of planar interface instability and the dynamical changes in pattern formation which lead to the selection of a steady-state cellular wavenumber is investigated. The timeevolution of a steady-state interface pattern shows that the actual wavelength selection process occurs in a highly nonlinear regime. The initial peak, formed in the linear regime, amplifies slowly compared to the peaks formed in the nonlinear regime. In the nonlinear regime, a series of wavenumber8 is observed which ultimately lead to a unique wavenumber peak as the system approaches the steady-state configuration. This final peak shows a significant width indicating that the wavelength selection criterion is not very sharp. Acknowledgments This work is supported by Ames Laboratory, which is operated for the U.S. Department of Energy by Iowa State University, under contract no. W-7405-Eng-82, supported by the Director of Energy Research, Office of Basic Energy Sciences. References 1.
2. 3. 4. 5. 6.
7. 8.
9. I0. ii. 12. 13. 14. 15. 16.
J. S. Langer, Rev. Mbd. Phys. 52, 1 (1980). See for examples several articles in: Physica 12D, 3-433 (1984). W. Kurz and D. J. Fisher, Solidification, Trans. Tech. Publ., Switzerland (1984). R. Trivedi, Metall. Trans. 15A, 977 (1984). K. Somhoonsuk, J. T. Mason and R. Trivedi, Metall. Trans. 15A, 967 (1984). R. Trivedi and K. Somboonsuk, J. Mat. Sci. Eng. 65, 65 (1984). S. de Cheveigne, C. Guthmann and M. M. Lebrun, J~-Cryst. Growth 73, 242 (1985). M. A. Eshelman, V. Seetharaman and R. Trivedi, Acta Metall. (in--press). B. Caroli, C. Caroli, B. Roulet and J. S. Langer, Phys. Rev. A 33, 442 (1986). M. Kerszberg, Physica 12D, 262 (1984). J. T. Mason and M. A. Eshelman, IS-4906, Ames L a b o r a t o r y , Ames, IA (1986). M. A. Eshelman, P h . D . T h e s i s , Iowa S t a t e U n i v e r s i t y (1987). R. T r i v e d i and K. Somboonsuk, Acta Metal1. 33, 1061 (1985). M. A. Eshelman, V. Seetharaman and R. T r i v e t , Unpublished work, Ames L a b o r a t o r y , Ames, IA 50011. W. W. Mullins and R. F. Sekerka, J. Appl. Phys. 35, 444 (1964). M.A. Eshelman and R. Trivedi, Acta Metall. 35, 2 ~ 3 (1987).
METALLURGICA
Vol. 22, pp. 8 9 3 - 8 9 8 , 1988 P r i n t e d in the U . S . A .
Pergamon
Press
plc
WAVELENGTH SELECTION OF CELLULAR PATTERNS M. A. Eshelman and R. Trivedi Ames Laboratory-USDOE and the Department of Materials Science and Engineering Iowa State University, Ames, IA 50011
(Received March 7, 1988)
Introduction The phenomenon of pattern formation in systems which are driven far away from thermodynamic equilibrium has received an increasing amount of theoretical interest in a variety of scientific disciplines [I-2]. One of the important systems, which we shall consider in this paper, is the directional solidification of a two-component system. In directional solidification, an alloy of fixed composition is solidified at a desired velocity under the condition of a constant positive temperature gradient in the liquid. At low velocities a planar solid-liquid interface exists. As the velocity is increased above some critical value, Vc, the planar interface becomes unstable and reorganizes into a periodic array of cells. When a planar interface is driven just beyond the critical velocity, Vc, experimental studies show that the the steady-state cellular structure which emerges has a definite wavelength and amplitude whose magnitudes depend on the value of the steady-state velocity [3-8]. Theoretical models, based on the solvability condition, however, predict a discrete set of wavelengths [9]. The principle which selects a specific wavelength from a set of possible wavelengths is not yet established, but it is generally believed that the selection process occurs in a highly nonlinear regime which allows the elimination or creation of cells to achieve the final spacing [8,10]. In order to obtain a better understanding of the dynamics of the wavelength selection process, we have carried out directional solidification studies in the succinonitrile - 0.15 wt.% acetone system. This system is transparent so that the dynamics of the pattern reorganization can be studied in situ. The changes in the interface shape with time were observed and analyzed by Fourier transforms. From the Fourier spectra, the dynamical changes in the pattern wavenambers with time were evaluated to determine the process by which the pattern selection occurs. Experimental Directional solidification studies were carried out in an apparatus which is described by Somboonsuk et al. [8] and Mason and Eshelman [Ii]. Both succinonitrile and acetone were first purified and then mixed under an inert environment to obtain a mixture with 0.15 wt.% acetone. The temperature gradient stage was thermally equilibrated to give a temperature gradient of 3.76 K/-,, at the interface. The system was directionally solidified at a velocity of 0.5 ~m/s for 210 min. During this time a steady-state planar interface was established. The velocity was then increased to 0.7 ~m/s for 18 min. and the interface was found to remain planar. The velocity was increased further to 0.8 ~m/s, where the planar interface became unstable after II min. The shape of the interface was photographed at 30 second intervals for 120 minutes. From the photomicrographs, the interface shape was digitized, and the resultant data set was input into a VAX computer for analysis. Figure 1 shows the changes in the interface shape as a function of time for a few selected times. The time at which the first distinguishable sign of instability occurred was taken to be t = O. The evolution of the interface pattern was analyzed in the wave vector space by Fourier transforming the digitized interface shapes for each time step. Within a small time range, some of the cellular perturbations folded back upon itself. The overhang of such cells was neglected, but the average location and the amplitude of these cells were not changed. In order to reduce
893 0036-9748/88 $3.00
+ .00
894
CELLULAR
3.5
i
i
i
i
PATTERNS
i
i
i
i
i
Vol.
i
22,
No.
6
i
5.0 ~
2.5 2.0 E E1
.
5
LO 0.5 0.5
1.0
1.5
2.0 2.5
mm
FIG. i.
Time evolution of an interface pattern in a stationary frame of reference. The interface profiles shown here are digitalized profiles of the experimental interface shapes observed in the directionally solidified succinonitrile - 0.15 at% acetone system at G ffi 3.76 k/mm and V = 0.8 ~m/s.
noise in the measurements, the input data set was smoothed with a three point center weighted smoothing function. The digitized data set, after the smoothing, was found to reproduce the interface shape quite accurately. Specific details of the Fourier analysis of interface shapes is given by Eshelman [12]. Experiments were also repeated to check the reproducibility of the results. Results Spatial Fourier transforms of patterns at times shortly after the occurrence of instability are shown in Fig. 2. The initial instability occurs with a peak at the waven~nber of 0.0708 ~m -I. There is also a peak at the wavent~aber of 0.003 ~m -I, but it is diffused due to the bias at the origin. This peak corresponds to the long wavelength perturbations which can be seen in Fig. I. The existence of two characteristic wavenumbers for the initial instability of the interface was also discussed by Trivedi and Somboonsuk [13]. Since the low wave number peak disappears in the cellular growth region once the interface is uniformly perturbed, we will not discuss it further. The general characteristics of the spectrum do not change significantly between 0 and 2 minutes, except that the peak at k = 0.0708 ~m -I shifts to the left, This shift corresponds to the slight increase in spacing which occurs between the pair of perturbations shown in Fig. I. In addition to the sh~ft, small peaks begin to appear at low wavenwaber values. A distinct change in the spectrum occurs between 2 and 3 minutes. We shall show later on that in this time range nonlinear effects become important. Figure 2b shows the Fourier spectra for t ffi 0, 5, i0, 15 and 20 minutes after the initial instability. Two important observations can be made: (I) a finite number of peaks is observed, and (2) the peak which corresponds to the lowest wavenumber begins to amplify at the largest rate. Figure 2c shows the spectra after 50 minutes. Note that one peak becomes prominent, indicating that the pattern is approaching the steady-state wavelength.
Vol.
22,
No.
6
CELLULAR
.10
PATTERNS
895
--t=Ornin - - - t = l min ...... t=2min ---t=3min ---t=4min
.08 ® .06.
~ .04 .00
'~,~
" ~ ji. ......
i/
.02
.00 .02 .04
.06 08 .I0 k,Fm"
]2
.14 ]6
.5 ..... t =Omin .... l : 5 min ....... t =lOmin - - - t=15min t=20min
A
.4
n
/I
:=
~'.Z
<~
.1
'.~.
..
..:
.
.0 .00 .02
.04
.06
.08
.10
]2
.14
.16
k,Fm-'
I0
(~
- - t= 50rain
II
2
% o o2
.o4 .o6 .o8 .~o .i2
.~4
]6
k,Fm" FIG.
2.
Spatial F o u r i e r t r a n s f o r m s of i n t e r f a c e p r o f i l e s taken at d i f f e r e n t time intervals.
896
CELLULAR
PATTERNS
Vol.
22,
No.
6
Discussion The spectral analysis of the interface shape gives insight into the dynamical nature of the pattern formation. The initial peak, which forms in the linear regime, amplifies more slowly compared to the subsequent peaks which form in the nonlinear regime. In order to examine the amplification rate of these various wavenumber peaks, the amplitude of the Fourier spectrum is plotted as a function of time for three characteristic peaks, as shown in Fig. 3. Three wavenumbers were chosen which represent the initial, intermediate and final wavenumber of the pattern. The initial peak, which corresponds to k = 0.0708 pm -I, shows a linear behavior up to about 2 minutes. After 2 minutes a sharp nonlinear behavior is observed. The amplitude of this peak begins to grow slowly as other peaks develop. The intermediate peak (k = 0.0507 Bm -I) amplifies at a very rapid rate and competes with the final peak until about 20 minutes. The final peak, which corresponds to k = 0.0199 pm -I becomes predominant only after 20 minutes. The final peak has a significant width (Fig. 2c) which shows that the wavelength selection criterion is not very sharp. The general characteristics of the wave number evolutions was found to be reproducible under different experimental conditions in this system as well as in the pivalic acid - ethanol system [14]. The pattern selection process can be divided into linear and nonlinear regions. In the linear region, one wavenumber predominates. Development of new peaks at lower wavenumber occurs when nonlinear effect become important. In the nonlinear regime, several peaks exist over a period of time. Since the differences in amplitudes of these peaks are small, it is not quite certain whether the middle three peaks are real or not. They could be represented by one wider peak. Several similar experiments, however, gave rise to similar peaks so that the existence of several intermediate peaks is indeed experimentally reproducible. At longer times, Fig. 2C, the lowest wavenumber peak begins to amplify sharply. This rapid amplification occurs because highly nonlinear effects cause cell elimination which allows the pattern to decrease the wavenumber
[8]. The wavenumber of the initial unstable pattern was found to be 0.07070 pm -I, which is somewhat larger the value of 0.017 pm -I predicted by the linear stability analysis of Mullins and Sekerta [15]. Since the amplication rate of the initial waven~nber is found to show a linear behaviour, fig. 3, one would expect the linear theory to predict the experimental observation. However, linear theory is based on the interface instability at a constant velocity. In our experiments, the initial instability was observed when the external velocity was changed from 0.7
1.0
-o- k:0.0199 ' --a-- k:O.0507 ---0.- k:0.0707
' .~
' ~
'/~'~'~t / " / [
=- 0.I o. E
0.01 O0
FIG. 3.
5.0
I0.0
15.0 Time,rain.
20.0
2.5.0
The variation in amplitude with time for selected wavenumbers. The wavenumber, k = 0.0199 pm -I, which has a smaller initial amplitude, becomes predominant at larger times when nonlinear effects become important.
Vol.
22,
No.
6
CELLULAR
PATTERNS
897
to 0.8 ~m/s. Since the experimental system is transparent, it was possible to measure the change in interface velocity with time and to identify the interface velocity at which the instability occurred. Figure 4 shows the variation in the interface velocity with time. In order to compare our results with the theory, it is important to note that the linear stability analysis is based on the steady-state solute profile in liquid. Since the velocity changes sharply in our experiments, dynamical effects become important which cause the concentration gradient at the interface to be larger than that predicted by the steady-state analysis. Eshelman and Trivedi [16] have examined this effect experimentally and shown that the dynamical results can be approximately compared with the steady-state model by making appropriate corrections in the velocity or composition value. These corrected velocity-wave nomber results are shown in Fig. 5 and compared with the theoretical prediction of Mullins and Sckerka [15]. The initial wavenumber could be measured only for the profile that was observed a few seconds after the break-up, at which time the velocity of the interface was slightly above Ve, as measured from Fig. 4. This result is shown as an open circle in Fig. 5, and it is close to the fastest growing wavenumber at that velocity. This experimental difficulty in observing the wavent~ber at the critical velocity may be responsible for experimental results which show initial w a v e n ~ b e r to be larger than that predicted by the linear stability analysis [7, 16]. Since the curvature of the V-k plot is very small at the critical velocity, a slight increase in velocity will significantly alter the wavenumber. As the velocity of the interface increases sharply with time, nonlinear effects become important which give rise to w a v e n ~ b e r peaks shown by crosses in Fig. 5. The final spacing, established after 50 minutes is shown as a filled circle. This spacing was found to continue to exist even when steady-state conditions were established. According to the linear theory, the planar interface should be stable at 0.8~m/s under steady-state conditions. However~ as shown by Eshelman and Trivedi [15], the planar to cellular transition in the succinonitrite-acetone system is subcritical so not the spacings established at V > V c will continue to be stable at V slightly below V c. These experiment results show that, as nonlinear effects become important, k decreases so that the time evolution of cellular pattern is accompanied by the coarsening of the initial pattern.
I
I
:::L
I
I
•
^,~,~E0.81"0 o.cj
I
I
I
I
I
I
I
•
_~
.
o
"~ 0.7 "E -0.6
I I I
lanor
0.5
FIG. 4.
I I
I
0
Nonplanor
I
I0
I
I
20 50
I
i
I
40 50 60 Time, min
70 80
The variations in interface velocity with time when the external velocity was changed from 0.7 to 0.8 ~m/s.
898
CELLULAR
I
I
I
Vol.
22,
No.
6
I
/
5•0
E
=L
_~ 2.o
w =
PATTERNS
o tnitiob wave n u m b e r
I.O
• Final wave n u m b e r x I n t e r m e d i a t e peak wave n u m b e r s I 0 4
LO-3
I I 0 "e
I I0 "1
I I
tO
Wave number, /~m" I
FIG. 5.
Relationship between velocity and waven,~ber, as predicted by the linear stability analysis, for succinonitrile - 0.15 wt% acetone alloys at G = 3.76 K/mm. The line between the experimental data prints show the velocity range over which the given wavenumber was predominant.
Conclusions The process of planar interface instability and the dynamical changes in pattern formation which lead to the selection of a steady-state cellular wavenumber is investigated. The timeevolution of a steady-state interface pattern shows that the actual wavelength selection process occurs in a highly nonlinear regime. The initial peak, formed in the linear regime, amplifies slowly compared to the peaks formed in the nonlinear regime. In the nonlinear regime, a series of wavenumber8 is observed which ultimately lead to a unique wavenumber peak as the system approaches the steady-state configuration. This final peak shows a significant width indicating that the wavelength selection criterion is not very sharp. Acknowledgments This work is supported by Ames Laboratory, which is operated for the U.S. Department of Energy by Iowa State University, under contract no. W-7405-Eng-82, supported by the Director of Energy Research, Office of Basic Energy Sciences. References 1.
2. 3. 4. 5. 6.
7. 8.
9. I0. ii. 12. 13. 14. 15. 16.
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