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Fluid patterns in the diffusive field around a growing crystal
j. . . . . . oF C R Y S T A L GROWTH
Journal of Crystal Growth 128 (1993) 163-166 North-Holland
Fluid patterns in the diffusive field around a growing crystal J u a n - M a G a r c l a - R u i z a n d F e r m l n Otfilora Instituto Andaluz de Geologia Mediterrdnea, CSIC, Universidad de Granada, Avda. Fuentenueua s/n, Granada 18002, Spain
T h e use of simultaneous diffusion of particles in a modified D L A system [Witten and Sander, Phys. Rev. Letters 47 (1981) 1400; Phys. Rev. B 27 (1983) 5686], leading to several morphologycal outputs, permitted us to observe the diffusive field around the growing cluster. We show that the existence of morphological instabilities produces the symmetry breaking of the diffusive field around the cluster and the formation of preferred flow induced by a morphological instability.
Although the morphological behaviour of crystals at equilibrium is rather well understood [1], the explanation of how they behave when growing apart from equilibrium conditions is yet obscure. The introduction of previous stochastic crystal growth simulation [2,3] to the recently developed diffusion limited aggregation (DLA) model [4,5] represents a new and powerful approach, as they avoid the analytical problems introduced by changing boundary conditions [6]. Previous studies have drawn the attention to the very cluster morphology, but as recently pointed out [7], the potential field around the cluster has been neglected. We have previously observed [8-10] that the morphological outputs of a DLA process can be tuned by allowing the aggregating particles to move along a diffusional path on the surface of the crystal, a process playing a crucial role in classical crystal growth theories [13,14]. The use in these simulations of a multi-particle diffusive field allows us to test the kinetics of the growth process and the evolution of the concentration pattern in the "fluid" surrounding the clusters. Diffusion field is simulated by allowing some particles (typically from 1000 to 5000) to move randomly to their neighbours whit equal probabilities. Boundary conditions imposed for computational effectiveness are: (a) a generation circle P where particles start their random walk, which is
centred in the cluster seed and is at least 50 pixels away from any point in the cluster surface, and (b) a perfect sink S with the same centre and a radius 10 pixels greater. The surface kinetics is simulated by allowing a surface walk of the particle for a time t s after its landing on the cluster surface (adatom mobility). For more details on the simulation, procedure, see ref. [8]. Although there are computer simulations describing crystal growth processes with higher sophistication [4,5], we have used this model as a simple way of tuning the morphological output of a crystal growth simulation keeping the sticking probability of the landing particles equal to one. The translation of our single parameter t~ in terms like bond strength or supersaturation can be easily done because t~ depends on the hop frequency for adatoms and the density of the two-dimensional nucleus which, in turn, are functions of the bond strength and supersaturation. Fig. 1 shows the effect obtained by using different values of t~. The value t s = 0 (fig. la) produces obviously the well-known fractal structure with d = 1.70 obtained in the classical DLA model [11,12]. By increasing ts (fig. lb), the result is a less ramified structure with wider branches. After further increases of t~ (fig. lc), anisotropy becomes evident and the symmetry P4mm (the one expected to emerge because of the bonding configuration of the particles) appears. Next, the
0022-0248/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
J.M. Garc[a-Ruiz, F. Otdlora / Fluid patterns in diffusive field around growing crystal
164
cluster adopt the morphology of a hopper crystal (fig. ld) and finally a perfect single crystal appears at very large t S (fig. le). Fig. 2 shows the concentration distribution at the end of the t s = 100 experiment (shown in fig. lc) in terms of isotones (grey and white ribbons). More information can be found in ref. [10]. Fig. 2 highlights some structures of the diffusion field which represent a breakdown of the radial symmetry of the field. Due to storage restrictions, our
Cellular pattern Generation Circle
Radial Profile Flux Path Tangential Profile
Fig. 2. Distribution of iso-concentration zones around a crystal growing under diffusional control (t s = 100). The isotones concentration was calculated for each pixel by counting the number of bulk diffusing particles inside a disk centred in this pixel and having a radius of 18 pixels. The figure shows the non-radial properties of the diffusion field (concentration profiles) and its influence in the apparition of curved flux paths and cellular patterns.
d
e
A
~
Fig. 1. Morphological behaviour of DLA clusters by using different surface diffusion time: (a) t~ = 0; (b) t~ = 10; (c) t s = 100; (d) t s = 1000; (e) t s = 10000.
result is static (we have solved the concentration field for the final growth stage). For this reason, we have made the following analysis to understand how these structures develop and whether they are real effects or artifacts introduced by boundary conditions. The formation of such structures was explored by sending random walkers from random positions on the generation circle P towards the cluster shown in fig. le. When a particle reaches the sink, it is removed and a new one is released from a random site on P. The same is done when a random walker reaches the cluster surface but, in this case, we store the launching and touching positions. This procedure was iterated until 1.8 x 105 particles reached the periphery of the cluster. The behaviour of the system is then studied by plotting the relevant properties versus the angular component of the polar coordinates of the launching (ae) a n d / o r touching (a t ) points. In fig. 3, we have plotted the mean deviations from perfect ballistic trajectories ~'= E ( a e a t ) / N , where N is the number of touching events inside the discretized interval represented by each point in the plot. ~" takes zero values in the direction of the centre of faces and along corners,
165
J.M. Garc[a-Ruiz, F. Ot6lora / Fluid patterns in diffusiue field around growing crystal
and positive or negative values d e p e n d i n g on the clockwise or counterclockwise position of the touching point referred to the centre of the face to which it belong. Those particles diffusing from the region facing the {11} surfaces will aggregate sooner than those diffusing from the region facing {10} surfaces. This feature creates a drop of the concentration in the region closer to preferred growth directions. This systematic distribution of trajectories provokes the symmetry breaking of the radial diffusive field and the existence of features resembling convection cells in the field. T h e behaviour of crystal faces during crystal growth, the onset of dendritic instabilities and the development of sidebranching should be related to this dynamic coupling between the diffusive field and the m o r p h o l o g y of the cluster growing from it. T h e effect of b o u n d a r y conditions can be easily u n d e r s t o o d by considering a one-dimensional analogue: in fig. 4 are represented probability distributions for a r a n d o m walker with different b o u n d a r y conditions. T h e results plotted are obtained as follows: a r a n d o m walker is initially set at x = 0 and allowed to move r a n d o m l y to the left or right during 500 time units. A f t e r this time its final position is recorded. T h e final positions of 105 of such r a n d o m walkers are used to c o m p u t e
4.8
3.6 2.4
~, 1.2 '-~ 0.0
-2.4
oo
~
o
~
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,~
~
~ ' ~
o
'°' L / . / ~
-6c
e~_
-7.;
°'/ ,
0.0
10,0
,
,n
20.0 30.0 40.0 50.0 60.0 70.0 Angular coordinate of the launching site
,
,
80.0
,
90.0
Fig. 3. Statistical properties of the diffusive field around the cluster shown in fig. le. A plot of the mean deviation (in degrees) from ballistic trajectories, i.e. difference between the average angular coordinate of the launching and touching point. Solid vertical lines limitates solid angles covered by the three faces involved. The enclosed box contains the reference system used to obtain data; arrows indicate the direction of deviations showed in the plot.
4000 3600 3200 ~2800 ~2400
'
J
°°
ql Clui~,rot 20
°°
2000
°°
1600
~
o °°
° ~° 1200
%,
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800 400 0 -100
".°
• JZ:I/ -80
-60
-40
*e -20
0
i 20
~°
40
60
80
100
x Fig. 4. Boundary conditions induced distortions of a one-dimensional diffusive field simulated by using random walkers. Probability distributions for random walkers after 500 jumps for 100,000 random walkers without any sink (circles), with a perfect sink at + 10 (squares), with perfect sinks at + 10 and 40 (triangles) and with perfect sinks at + 10 and - 20. -
the probability distribution. T h e growing crystal is assumed to be in the negative direction, the generation circle at x = 0 and the perfect sink in the positive direction. T h e first b o u n d a r y condition is the existence of a generation circle; this is simulated by setting all the r a n d o m walkers at x = 0 for t = 0. As a result, the diffusion field displays a mirror symmetry (circles in fig. 4), i.e., it decreases in the direction of the cluster (negative direction of 4) as well as in the positive direction ( h o m o g e n e o u s infinite). This has no influence for all practical purposes. Simply the field is neglected b e y o n d the generation circle). If now a sink circle is imposed (squares in fig. 4, sink at + 10), the distribution shifts, its maxim u m is no longer at x = 0 and the net probability is lower at areas closer to the generation circle. This explains the shift of maximum concentrations in fig. 2 toward the crystal. N o t e that, for distances greater than 50 between the most protrudent tip and the generation circle (as in our simulation), the ideal and real curves are almost identical. A new b o u n d a r y condition is set if now the growing crystal acts itself as a sink for particles. N o t e that this third b o u n d a r y condition is not imposed for computational effectiveness but is a real fact. Setting a second sink at x = - 40 (triangles) and at x = - 2 0 (diamonds), the result is to obtain steeper gradients and a minor shift of the
166
J.M. Garda-Ruiz, F. Otdlora /Fluidpatterns in diffusive fieM around growing crystal
maximum and net field. T h e steeper gradients are a logical c o n s e q u e n c e of the crystal-field interplay and show the origin of the Berg effect. The shifts in the position and values are responsible of the a p p e a r a n c e of cellular patterns. O u r a p p r o a c h to the c o m p u t a t i o n of the diffusional field a r o u n d a growing cluster is complementary to approaches such as that of Mandelbrot and Everstz [7], because it is known that the probability distribution of a r a n d o m walker fits Fick's second law. So, at least partly, some comm o n information is expected to be obtained from both approaches. W a t h e v e r the m e t h o d used to solve the concentration field, it is clear that as soon as the growing crystal shows its characteristic anisotropy, the field b e c o m e s non-radial. This inhomogeneity in the field is enlarged as the crystal develops corners, protrusions and, finally, dendrites. Preferred flow lines start to appear, of which the origin and evolution are closely linked to the anisotropy of the crystal and its growth behaviour. As described above, our algorithm simulates a transport-limited growth because surface kinetics is instantaneous. In this case, the supply of nutrient varies due to preferred flux directions in the diffusion field, which produces an amplification of the growth rate of instabilities that, in turn, modify the preferred flux directions. A n additional result of this work is to show the importance of a careful design and interpretation of crystal growth simulations. To sketch real world
in a simulation always produces a mixing of real effects and artifacts. A l t h o u g h further work is n e e d e d to understand this mixing, a a d e q u a t e filter can isolate interesting results. W o r k p e r f o r m e d u n d e r financial support from the A u t o n o m o u s G o v e r n m e n t of A n d a l u c l a (Spain) and C I C Y T Project PB89-0059.
References [1] P. Hartman and W.G. Perdok, Acta Cryst. 8 (1955) 521. [2] G.H. Gilmer, J. Crystal Growth 42 (1977) 3. [3] G.H. Gilmer, Science 208 (1980) 355. [4] Y. Saito and T. Ueda, Phys. Rev. A 40 (1989) 3408. [5] R.F. Xiao, J.I.D. Alexander and F. Rosenberger, Phys. Rev. A 38 (1988) 2447. [6] J.S. Langer, Rev. Mod. Phys. 52 (1980) 1. [7] B.B. Mandelbrot and C.J.G. Evertsz, Nature 348 (1990) 143. [8] J.M. Garcla-Ruiz and F. Ot~ilora, Physica A 178 (1991) 415. [9] J.M. Garcla-Ruiz and F. Otfilora, J. Crystal Growth 118 (1992) 160. [10] F. Ot~lora and J.M. Garcla-Ruiz, in: Crystal Growth, Proc. 3rd European Conf., Ed. A. L6rinczy (Trans Tech, Zurich, 1991). [11] T.A. Witten and L. Sander, Phys. Rev. Letters 47 (1981) 1400. [12] T.A. Witten and L. Sander, Phys. Rev. B 27 (1983) 5686. [13] W.K. Burton, N. Cabrera and C.F. Frank, Phil. Trans. Roy. Soc. London 243 A (1951) 299. [14] A.A. Chernov, Kristallografiya 8 (1963) 87.
Journal of Crystal Growth 128 (1993) 163-166 North-Holland
Fluid patterns in the diffusive field around a growing crystal J u a n - M a G a r c l a - R u i z a n d F e r m l n Otfilora Instituto Andaluz de Geologia Mediterrdnea, CSIC, Universidad de Granada, Avda. Fuentenueua s/n, Granada 18002, Spain
T h e use of simultaneous diffusion of particles in a modified D L A system [Witten and Sander, Phys. Rev. Letters 47 (1981) 1400; Phys. Rev. B 27 (1983) 5686], leading to several morphologycal outputs, permitted us to observe the diffusive field around the growing cluster. We show that the existence of morphological instabilities produces the symmetry breaking of the diffusive field around the cluster and the formation of preferred flow induced by a morphological instability.
Although the morphological behaviour of crystals at equilibrium is rather well understood [1], the explanation of how they behave when growing apart from equilibrium conditions is yet obscure. The introduction of previous stochastic crystal growth simulation [2,3] to the recently developed diffusion limited aggregation (DLA) model [4,5] represents a new and powerful approach, as they avoid the analytical problems introduced by changing boundary conditions [6]. Previous studies have drawn the attention to the very cluster morphology, but as recently pointed out [7], the potential field around the cluster has been neglected. We have previously observed [8-10] that the morphological outputs of a DLA process can be tuned by allowing the aggregating particles to move along a diffusional path on the surface of the crystal, a process playing a crucial role in classical crystal growth theories [13,14]. The use in these simulations of a multi-particle diffusive field allows us to test the kinetics of the growth process and the evolution of the concentration pattern in the "fluid" surrounding the clusters. Diffusion field is simulated by allowing some particles (typically from 1000 to 5000) to move randomly to their neighbours whit equal probabilities. Boundary conditions imposed for computational effectiveness are: (a) a generation circle P where particles start their random walk, which is
centred in the cluster seed and is at least 50 pixels away from any point in the cluster surface, and (b) a perfect sink S with the same centre and a radius 10 pixels greater. The surface kinetics is simulated by allowing a surface walk of the particle for a time t s after its landing on the cluster surface (adatom mobility). For more details on the simulation, procedure, see ref. [8]. Although there are computer simulations describing crystal growth processes with higher sophistication [4,5], we have used this model as a simple way of tuning the morphological output of a crystal growth simulation keeping the sticking probability of the landing particles equal to one. The translation of our single parameter t~ in terms like bond strength or supersaturation can be easily done because t~ depends on the hop frequency for adatoms and the density of the two-dimensional nucleus which, in turn, are functions of the bond strength and supersaturation. Fig. 1 shows the effect obtained by using different values of t~. The value t s = 0 (fig. la) produces obviously the well-known fractal structure with d = 1.70 obtained in the classical DLA model [11,12]. By increasing ts (fig. lb), the result is a less ramified structure with wider branches. After further increases of t~ (fig. lc), anisotropy becomes evident and the symmetry P4mm (the one expected to emerge because of the bonding configuration of the particles) appears. Next, the
0022-0248/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
J.M. Garc[a-Ruiz, F. Otdlora / Fluid patterns in diffusive field around growing crystal
164
cluster adopt the morphology of a hopper crystal (fig. ld) and finally a perfect single crystal appears at very large t S (fig. le). Fig. 2 shows the concentration distribution at the end of the t s = 100 experiment (shown in fig. lc) in terms of isotones (grey and white ribbons). More information can be found in ref. [10]. Fig. 2 highlights some structures of the diffusion field which represent a breakdown of the radial symmetry of the field. Due to storage restrictions, our
Cellular pattern Generation Circle
Radial Profile Flux Path Tangential Profile
Fig. 2. Distribution of iso-concentration zones around a crystal growing under diffusional control (t s = 100). The isotones concentration was calculated for each pixel by counting the number of bulk diffusing particles inside a disk centred in this pixel and having a radius of 18 pixels. The figure shows the non-radial properties of the diffusion field (concentration profiles) and its influence in the apparition of curved flux paths and cellular patterns.
d
e
A
~
Fig. 1. Morphological behaviour of DLA clusters by using different surface diffusion time: (a) t~ = 0; (b) t~ = 10; (c) t s = 100; (d) t s = 1000; (e) t s = 10000.
result is static (we have solved the concentration field for the final growth stage). For this reason, we have made the following analysis to understand how these structures develop and whether they are real effects or artifacts introduced by boundary conditions. The formation of such structures was explored by sending random walkers from random positions on the generation circle P towards the cluster shown in fig. le. When a particle reaches the sink, it is removed and a new one is released from a random site on P. The same is done when a random walker reaches the cluster surface but, in this case, we store the launching and touching positions. This procedure was iterated until 1.8 x 105 particles reached the periphery of the cluster. The behaviour of the system is then studied by plotting the relevant properties versus the angular component of the polar coordinates of the launching (ae) a n d / o r touching (a t ) points. In fig. 3, we have plotted the mean deviations from perfect ballistic trajectories ~'= E ( a e a t ) / N , where N is the number of touching events inside the discretized interval represented by each point in the plot. ~" takes zero values in the direction of the centre of faces and along corners,
165
J.M. Garc[a-Ruiz, F. Ot6lora / Fluid patterns in diffusiue field around growing crystal
and positive or negative values d e p e n d i n g on the clockwise or counterclockwise position of the touching point referred to the centre of the face to which it belong. Those particles diffusing from the region facing the {11} surfaces will aggregate sooner than those diffusing from the region facing {10} surfaces. This feature creates a drop of the concentration in the region closer to preferred growth directions. This systematic distribution of trajectories provokes the symmetry breaking of the radial diffusive field and the existence of features resembling convection cells in the field. T h e behaviour of crystal faces during crystal growth, the onset of dendritic instabilities and the development of sidebranching should be related to this dynamic coupling between the diffusive field and the m o r p h o l o g y of the cluster growing from it. T h e effect of b o u n d a r y conditions can be easily u n d e r s t o o d by considering a one-dimensional analogue: in fig. 4 are represented probability distributions for a r a n d o m walker with different b o u n d a r y conditions. T h e results plotted are obtained as follows: a r a n d o m walker is initially set at x = 0 and allowed to move r a n d o m l y to the left or right during 500 time units. A f t e r this time its final position is recorded. T h e final positions of 105 of such r a n d o m walkers are used to c o m p u t e
4.8
3.6 2.4
~, 1.2 '-~ 0.0
-2.4
oo
~
o
~
G
o° o
,~
~
~ ' ~
o
'°' L / . / ~
-6c
e~_
-7.;
°'/ ,
0.0
10,0
,
,n
20.0 30.0 40.0 50.0 60.0 70.0 Angular coordinate of the launching site
,
,
80.0
,
90.0
Fig. 3. Statistical properties of the diffusive field around the cluster shown in fig. le. A plot of the mean deviation (in degrees) from ballistic trajectories, i.e. difference between the average angular coordinate of the launching and touching point. Solid vertical lines limitates solid angles covered by the three faces involved. The enclosed box contains the reference system used to obtain data; arrows indicate the direction of deviations showed in the plot.
4000 3600 3200 ~2800 ~2400
'
J
°°
ql Clui~,rot 20
°°
2000
°°
1600
~
o °°
° ~° 1200
%,
°.
800 400 0 -100
".°
• JZ:I/ -80
-60
-40
*e -20
0
i 20
~°
40
60
80
100
x Fig. 4. Boundary conditions induced distortions of a one-dimensional diffusive field simulated by using random walkers. Probability distributions for random walkers after 500 jumps for 100,000 random walkers without any sink (circles), with a perfect sink at + 10 (squares), with perfect sinks at + 10 and 40 (triangles) and with perfect sinks at + 10 and - 20. -
the probability distribution. T h e growing crystal is assumed to be in the negative direction, the generation circle at x = 0 and the perfect sink in the positive direction. T h e first b o u n d a r y condition is the existence of a generation circle; this is simulated by setting all the r a n d o m walkers at x = 0 for t = 0. As a result, the diffusion field displays a mirror symmetry (circles in fig. 4), i.e., it decreases in the direction of the cluster (negative direction of 4) as well as in the positive direction ( h o m o g e n e o u s infinite). This has no influence for all practical purposes. Simply the field is neglected b e y o n d the generation circle). If now a sink circle is imposed (squares in fig. 4, sink at + 10), the distribution shifts, its maxim u m is no longer at x = 0 and the net probability is lower at areas closer to the generation circle. This explains the shift of maximum concentrations in fig. 2 toward the crystal. N o t e that, for distances greater than 50 between the most protrudent tip and the generation circle (as in our simulation), the ideal and real curves are almost identical. A new b o u n d a r y condition is set if now the growing crystal acts itself as a sink for particles. N o t e that this third b o u n d a r y condition is not imposed for computational effectiveness but is a real fact. Setting a second sink at x = - 40 (triangles) and at x = - 2 0 (diamonds), the result is to obtain steeper gradients and a minor shift of the
166
J.M. Garda-Ruiz, F. Otdlora /Fluidpatterns in diffusive fieM around growing crystal
maximum and net field. T h e steeper gradients are a logical c o n s e q u e n c e of the crystal-field interplay and show the origin of the Berg effect. The shifts in the position and values are responsible of the a p p e a r a n c e of cellular patterns. O u r a p p r o a c h to the c o m p u t a t i o n of the diffusional field a r o u n d a growing cluster is complementary to approaches such as that of Mandelbrot and Everstz [7], because it is known that the probability distribution of a r a n d o m walker fits Fick's second law. So, at least partly, some comm o n information is expected to be obtained from both approaches. W a t h e v e r the m e t h o d used to solve the concentration field, it is clear that as soon as the growing crystal shows its characteristic anisotropy, the field b e c o m e s non-radial. This inhomogeneity in the field is enlarged as the crystal develops corners, protrusions and, finally, dendrites. Preferred flow lines start to appear, of which the origin and evolution are closely linked to the anisotropy of the crystal and its growth behaviour. As described above, our algorithm simulates a transport-limited growth because surface kinetics is instantaneous. In this case, the supply of nutrient varies due to preferred flux directions in the diffusion field, which produces an amplification of the growth rate of instabilities that, in turn, modify the preferred flux directions. A n additional result of this work is to show the importance of a careful design and interpretation of crystal growth simulations. To sketch real world
in a simulation always produces a mixing of real effects and artifacts. A l t h o u g h further work is n e e d e d to understand this mixing, a a d e q u a t e filter can isolate interesting results. W o r k p e r f o r m e d u n d e r financial support from the A u t o n o m o u s G o v e r n m e n t of A n d a l u c l a (Spain) and C I C Y T Project PB89-0059.
References [1] P. Hartman and W.G. Perdok, Acta Cryst. 8 (1955) 521. [2] G.H. Gilmer, J. Crystal Growth 42 (1977) 3. [3] G.H. Gilmer, Science 208 (1980) 355. [4] Y. Saito and T. Ueda, Phys. Rev. A 40 (1989) 3408. [5] R.F. Xiao, J.I.D. Alexander and F. Rosenberger, Phys. Rev. A 38 (1988) 2447. [6] J.S. Langer, Rev. Mod. Phys. 52 (1980) 1. [7] B.B. Mandelbrot and C.J.G. Evertsz, Nature 348 (1990) 143. [8] J.M. Garcla-Ruiz and F. Ot~ilora, Physica A 178 (1991) 415. [9] J.M. Garcla-Ruiz and F. Otfilora, J. Crystal Growth 118 (1992) 160. [10] F. Ot~lora and J.M. Garcla-Ruiz, in: Crystal Growth, Proc. 3rd European Conf., Ed. A. L6rinczy (Trans Tech, Zurich, 1991). [11] T.A. Witten and L. Sander, Phys. Rev. Letters 47 (1981) 1400. [12] T.A. Witten and L. Sander, Phys. Rev. B 27 (1983) 5686. [13] W.K. Burton, N. Cabrera and C.F. Frank, Phil. Trans. Roy. Soc. London 243 A (1951) 299. [14] A.A. Chernov, Kristallografiya 8 (1963) 87.