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Monomer aggregation model for DLA dendrite crossover
PHYSiCA ELSEVIER
Physica A 224 (1996) 412-421
Monomer aggregation model for DLA dendrite crossover G. Ananthakrishna, Silvester J. Noronha Materials Research Centre, Indian Institute of Science, Bangalore 560 012, India
Abstract We consider Monte Carlo simulation of monomer aggregation into clusters with a view to simulate both DLA and dendrites. We introduce a new method of suppressing fluctuations by allowing two types of relaxations. First is a local on-site relaxation of monomers and the second, diffusive relaxation of the monomers along the perimeter of the cluster. The former is characterized by two energies and the latter by two activation energies corresponding to diffusion along edges and diffusion motion which is transverse locally. The model produces both DLA- and dendritelike patterns for different choices of parameters. We find two distinct configurations of dendrites corresponding to two distinct anisotropic directions.
I. Introduction Patterns formed under non-equilibrium growth conditions have been an object of attention for a long time [ 1]. These include a rich variety of dendritic patterns which appear to be very regular, in addition to the randomly ramified structures [2,3]. Such patterns have been observed in physical, chemical and biological systems [3,4]. Typically, the growth occurs by transport of heat or material by diffusion. Therefore in both cases, solution of the Laplacian plays an important role. The main aim of theoretical models is to understand and reproduce these patterns. There are essentially two distinct approaches to these problems. The first one is basically a continuum deterministic approach wherein the essential physics is described by a heat conduction equation, the interfacial moving boundary condition and inclusion of surface tension effects [ 1,6]. Different types of patterns result basically from the competition of the enhanced growth due to non-local diffusion field of that part of the crystal which bulges into the melt and the surface tension effects which tend to flatten the interface. Numerical solution Elsevier Science B.V. SSD10378-4371 (95)00359-2
G. Ananthakrishna, S.J. Noronha/Physica A 224 (1996) 412-421
413
of these equations supplemented with an equation for the environmental factors already gives patterns that are very close to real snowflakes [5]. The anisotropic direction that appears to manifest is either a consequence of the surface tension anisotropy or that of the underlying lattice on which the Laplacian is solved. In the last few years there is an improved understanding of the various morphologies arising given the anisotropy and the undercooling [6]. In contrast, the usual DLA model which proceeds by adding monomers to the cluster in an irreversible way produces ramified patterns [2,3] which do not posses any symmetry. Even though, the latter is known be related to the growth of regular dendrites [2], it is not related in any obvious way. The later method can be termed as 'discrete', since it attempts to build the cluster by irreversible aggregation of monomers at a time. While the first method is suitable for the description of dendritic crystal, the second produces ramified patterns. Both these approaches have given a good deal of understanding of the formation of the patterns. Fluctuations appear to be suppressed in the case of dendrites while in the case of the DLA, fluctuations dominate. Since these two approaches start from different ends, there have been attempts to synthesize an intermediate approach which starts with DLA approach but attempts to suppress the fluctuations [7-13]. One of the first methods was to include relaxation of the particle to lowest energy configuration in the neighbourhood of the point where the particle arrives. They include an algorithm to mimic the surface tension effects [8,9]. One method which has been very successful in producing the crossover behaviour from DLA to dendrites, involves suppressing the fluctuations by using a counter which keeps track of the number of random walkers visiting a site on the perimeter [10-13]. A monomer is added once the counter exceeds a certain value m*. The limit m* --+ 1 produces the DLA, while m* ~ c~ produces compact clusters. Suppressing fluctuations coupled with using the solution of the Laplacian as in the dielectric breakdown model appears to produce realistic snowflake structures [ 11-13]. Again, in these studies the anisotropy arises due to the lattice structure. In such models as above, the effect of changes in the experimentally controlled parameters such as temperature, extent of supersaturation or undercooling cannot be easily understood. It is the purpose of the present work to come up with a model wherein the theoretical parameters have a direct correspondence with the experimental parameters. In this brief study we include only temperature related parameters. Even within the limited scope of the present model, we find both DLA and dendritic type of clusters. Surprisingly, we find that the morphology of dense branched dendrites picks out two equivalent anisotropic directions, one of which is natural to the nucleus and the other to the line joining the corners.
2. The model Consider a monomer performing an off-lattice random walk in two dimensions. Each monomer is considered to have a finite size and is assumed for the sake of convenience to be of square shape with an edge length denoted by 8. (Other shapes can also be con-
414
G. Ananthakrishna, S..I. Noronha/Physica A 224 (1996) 412-421
(a)
(b) i .....
-.'. . . . . . . . . . .
' ...............
,~ . . . . . . . . .
J ...........
J
.........
i ....
J
i
i
! ' !l
Fig. 1. (a) and (b) refer to the two differentinterpretations of the on-site relaxation. Only (b) is relevant for the model for which the results are presented. However,no spatial relaxation is carried out in the simulation. sidered.) As in any DLA simulation, the monomers are released from the circumference of the circle whose radius is kept large. We do not include any orientational dynamics while the monomers are diffusing towards the aggregate, i.e., the edges of the monomers remain parallel to the initial directions of the released monomers. A single monomer located at the origin (with its center of mass coinciding with the origin) with its edges parallel to the x - y axes acts as the nucleus. In this model we allow the aggregation of the monomer such that the edges of the monomer match with the nucleus (or any of the free edges of a monomer which is part of the aggregate). We accomplish this in the following way. Consider a monomer reaching the neighbourhood of the nucleus within 26 (measured from the respective center of masses of the particles involved) for the first time. Since the diffusive motion of the monomer is an off-lattice random walk, the distance between the center of mass of the nucleus and the monomer will not be in general an integer multiple of the edge length. From conventional molecular dynamics calculations on any aggregation process, we know that particles move in a potential which is a sum of two body potentials depending on the distances between the particles. Only when they occupy the minimum potential energy configuration the clusters are considered stable. We believe that there is a similar situation in the present aggregation process also. Thus, we assume that once the monomer arrives for the first time within a distance 26 from the nucleus, it experiences a force of attraction which increases as a function of its distance, reaching a maximum when the distance between them is 6. We refer to this as on-site relaxation. This in principal involves both rotational relaxation as well as changes in the distance between the center of mass of the nucleus and the monomer such that the edges eventually become parallel and match. This is shown in Figs. l a,b. Preliminary simulations which were initially carried out showed that these simulations were very time consuming. Therefore, we have simplified this and several other aspects o f the above model as stated below: a) We allow the aggregation to proceed by matching of the edges of the monomers with the monomers belonging to the perimeter sites of the aggregate. b) The first simplification effected is that the monomers are released from the circumference of the circle with the edges of the monomers always parallel to the nucleus. Since we do not have any orientational dynamics in the diffusion of the monomers, they always arrive in the neighbourhood (see below) of the nucleus with their edges parallel to the edges of the nucleus. Even as the aggregation proceeds,
G. Ananthakrishna, S.J. Noronha/Physica A 224 (1996) 412-421
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the monomers arrive with their edges parallel to the monomers at the perimeter sites of the aggregate. c) The perimeter sites of the aggregate are defined as those sites which are not interior sites of the aggregate and therefore have less than three monomers (belonging to the aggregate) as their neighbours, (Note that in the aggregate, the centers of mass of the neighbours are all at distance 8.) Once the monomer arrives at a distance within 28 from any perimeter site (it could be more than one perimeter site) we consider that it is subject to a force of attraction as stated above. However, we do not explicitly carry out the movement of the monomer, but we keep track of the changes in the energy only, as a function of time. This is adequate for the present Monte-Carlo calculation since we need to effect the addition of the monomer at the site of the aggregate only when the monomer has a sticking probability equal to unity. The time-dependent binding energy is taken to be bn(t) = bn(0) + d[1 - e x p ( - t / ~ ' ) ].
(i)
Here, bn(t) is the binding energy of the monomer at the nth site at time t, bn(O) is the binding energy at time t = 0 (corresponding to a distance 8 from the perimeter site), ~- is the time scale of relaxation and A + bn(O) refers to the limiting binding energy of the bond established by the monomer with the cluster on sticking. (See Fig. lb.) d) The number of bonds established by the monomer is taken to be equal to the number of distances from the center of mass of the monomer to all possible perimeter site monomers of the aggregate which are less than 28. We further assume that all of them have the same value as long as these distances are less than 28. Corresponding to a fiat interface, there is only one bond (Fig. 2a). If there is a step as in Fig. 2b, the monomer can have two bonds. We take the particle to have three bonds corresponding to a well (Fig. 2c). e) The monomer can either stay at the site and relax on-site if it does not hop to its neighbouring sites, or it can hop to the neighbouring sites. Note that t refers to the time for relaxation at that site. If the particle moves to a neighbouring site, the time for the on-site relaxation starts at zero for that site. When the particle arrives within 28 length of the aggregate, it can also diffuse along the perimeter sites with a probability of migration proportional to e x p ( - E m / k T ) . We introduce two distinct energies of migration, one corresponding to migration on a fiat interface, E,(e) (Fig. 2a) and another to migration around a sharp corner, E(mr) (Figs. 2d,e). These two physically correspond to diffusive motion parallel and perpendicular to any branch of the aggregate. Monte Carlo simulation of the aggregation is carded out by the conventional energy minimization. The probability of hopping from the nth to the (n 4- 1 )th site is given by Pn,n+l O~ e -(E'+ae','±t)/kT",
(2)
416
G. Ananthakrishna, S.J. Noronha / Physica A 224 (1996) 412-421
(b)
(a) i.......... i ~......... d
1...........l...........!.........l................. t..........l.............'i
col
(0)
i.........J
...........J
tf!ll'i ~
i
,
i
1
i
[
.....
I
(f)
(e)
...........
I ~!
r
J ~
,iii[
Fig. 2. Distinct types of configurations of the monomer with the aggregate.
where E m = E,~e) or E~r) depending on the position of the particle. AEn,n+l = b , ( 0 ) + A [ l - e x p ( - t / r ) ] - b , ± l (0) is the change in the energy between the initial and the final states. Fig. 2a corresponds to migration along flat edges and Figs. 2d or 2e correspond to diffusive motion around sharp corners which locally is transverse to the motion along the edge. The probability of staying at the site ps = 1 - P,,,-1 - Pn.n+l. Due to the fact that there is an on-site time-dependent relaxation, we consider the particle to stick to a perimeter site if the on-site staying probability is more than a certain fixed number p*. This parameter has been essentially introduced to minimize the computational time. In our calculations this number is chosen to be between 0.8 to 0.95. Figs. 2a to 2e show typical configurations that are possible. From Eq. (2) it is clear that in the case of Fig. 2a, the monomer can move to the left or to the right with equal probability as it arrives at a distance 6 from the cluster (i.e. at t = 0). If it does not hop to the neighbouring site, then it is clear that the probability of staying at the site increases as a function of time. Further, at any later time, if the particle hops to a neighbouring site, we assume that the monomer hops to the least relaxed state. Consider the situation shown in Fig. 2f. From Eq. (2), the monomer has a higher probability to
G. Ananthakrishna, S.J. Noronha/ Physica A 224 (1996) 412-421
Fig. 3. DLA structure produced for
E~)IkT = E(mr)/kT = 4, bn(O)/kT
= 1.0,
A/kT
417
= 2.0, p * = 0.8 and
7"=1.
move to the left than to the right. In addition, having arrived there (Fig. 2b), it has a greater probability to eventually stick there. Consider the situation shown in Figs. 2d and 2e. The energy of migration for this situation is E~r~. The possible values E~mr~ can take depend on the actual physical situation. For our purposes we consider all possibilities, E(me) > E (r) , E(me) = E~~ and E (e) < E (e). Note that while E~m~ governs the diffusive motion of the particle along the branch of the aggregate, E~mr~ refers to the local transverse motion of the aggregate. Note, also Fig. 2c corresponds to the lowest energy configuration.
3. Results and discussion
All energies we use are in scaled units of kT. It is clear that when both the migration energies E ~e~ and E~,~ are large, we should expect that the particle has very little tendency to diffuse along the perimeter of the aggregate. Therefore, we should see a DLA-like structure when a binding energy, b0 and A are reasonable. Fig. 3 shows a typical DLA-like structure. The effects of A and bn (0) are interesting. If A is small, the on-site relaxation is not significant. If now the energy of migration E~e~ is decreased favouring diffusion along the fiat portion of the perimeter of the aggregate, thick branches can arise, if the bond energy bn(O) is reasonable. This is because the monomers have a tendency to aggregate at steps (as in Fig. 2c). A typical such densely branched morphology is shown in Fig. 4. In this case, the value of E~r~ chosen is unfavourable for the monomers to diffuse round sharp corners. This leads to no preferential growth direction (Fig. 4). If we further allow a decrease in E~e~, it favours rapid diffusion along the fiat parts of the aggregate, if in addition we choose E~r~ to be large, the monomer has
418
G. Ananthakrishna, S.J. Noronha/Physica A 224 (1996) 412-421
Fig. 4. Dense branched nearly symmetric pattern produced for E~me)/kT= 0.5, E~mr)/kT= 2, bn(O)/kT = 1.0, zl/kT = 0.05, p* = 0.8 and ~"= 1.
Fig. 5. Dendrite type morphology produced with x-y anisotropic direction for E(e)/kT = 0.01, E~r)/kT = 3, bn(O)/kt = 1.0, zl/kT = 0.05, p* = 0.8 and T = I. very little probability o f m o v e m e n t round the corners. This aids the growth of the cluster in the x - y directions, thereby p i c k i n g out the anisotropic growth directions seen in Fig. 5. This direction essentially corresponds to the natural anisotropic direction o f the nucleus at the origin. C o n s i d e r n o w a situation wherein both migration energies, corresponding to diffusion along and perpendicular to the branch are chosen to be o f the same order o f magnitude. A typical such figure is shown in Fig. 6. The aggregate has a dense branched m o r p h o l o g y typical to many dendritic crystals with an anisotropic direction at 45 ° to the orignal nucleus. This anisotropic directionality is surprising considering
the fact that the nucleus does not posses this symmetry. Clearly, this direction has been
G. Ananthakrishna, S.J. Noronha/ Physica A 224 (1996) 412-421
419
Fig. 6. Thick branched dendrite type morphology produced at 45° to the nucleus along the x-y. E~e)/kT = E~)/kT = 1, bn(O)/kT = 1.0, A/kT = 2.0, p* = 0.8 and r = 1.
Fig. 7. Dendritic structure produced for E~e)/kT = 0.5, E~mr)/kT= 3, bn(O)/kT = 0.5, zl/kT = 0.5, p* = 0.8 and r = 8.
picked out due to the competition between two directions of diffusion. It must be noted that in this case, the thickness o f the branches are much more than in the case o f Fig. 5. The effect o f the relaxation time r also has an important effect on the morphology of the aggregate. For the set o f parameters when the growth is preferentially in the x - y directions, increasing 7- gives rise to several branches with the aggregate having nearly a square shape. This is shown in Fig. 7. Lastly, the effect of increasing the bn(0) is to make the branches thicker as shown in Fig. 8. As mentioned earlier, the model in its present form does not include other effects such as the flux o f the incoming particles etc. Refinements of the model to various physical situations are under way. Here, we would like to emphasize that our attempt is not so much to produce realistic snowflake-like patterns, but to examine how the crossover arises as a function o f various physical parameters. Even so, the preliminary results are sufficiently encouraging that we expect that this model with appropriate modifications should be able to describe some physical situations. For instance, we expect that once the
420
G. Ananthakrishna, S.J. Noronha/Physica A 224 (1996) 412-421
Fig. 8. Thick branched dendrite structure produced for E(me)/kT = 0.5, E~r)/kT = 3.0, bn(O)/kT = 2.0, A/kT = 0.5, p* = 0.8 and ~"= 1. flux rate and circular geometry o f the particles are included, the model should be able to describe the recent results on the growth o f A g clusters in molecular vapour deposition on Pt (111) or A g (111) substrate [14,15]. Another instance we can visualize is a situation wherein the monomers are diffusing in a solution and aggregating to form different types o f patterns such as the D L A patterns and dendrites. Indeed, such a situation has been recently seen in the aggregation o f gold monomers [ 16]. We are in the process o f refining the model to the actual physical situation. In conclusion, in this model, we have devised an alternate method o f suppressing fluctuations which is physically closer to real situations. This has been accomplished by allowing for the diffusive relaxation coupled with energy minimization. This helps to sample a large number o f energy states. The D L A limit in this model arises when the migration energies are high and hence does not allow for suppression o f fluctuations. A cross over to dendritic type structure having two distinct anisotropic directions is shown to arise as a consequence o f competition o f two different energies o f migration. When diffusion in the transverse direction is prevented (E{me) << E{mr}), the dendrite picks out the natural anisotropic direction of the nucleus. In contrast, when both transverse and parallel directions o f diffusion compete, a new anisotropic direction which is not natural to the monomer is picked up. Further, the model has parameters which are experimentally accessible and have easy physical interpretation.
References [ 1] [21 [3] [4] [5] I6]
J.S. Langer, Rev. Mod. Phys. 50 (1980) 1, and references therein. T.A. Witten and L.M. Sander, Phys. Rev. Lett. 47 (1981) 1400. T. Vicsek, Fractal Growth Phenomenon (World Scientific, Singapore, 1992). J. Feder, Fractals (Plenum, New York, 1988). E Family, D.E. Platt and T. Vicsek, J. Phys. A 20 (1987) Ll177. T. lhle and H. Muller-Krumbhaar,Phys. Rev. E 49 (1994) 2972; Phys. Rev. Lett. 70 (1993) 3083, and references therein.
G. Ananthakrishna, S.J. Noronha/ Physica A 224 (1996) 412-421 [71 [8] [9] [10] [lll [12] [13] [14] [15] 116]
421
J. Szep, J. Cserti and J. Kertesz, J. Phys. A 18 (1985) IA13. T. Vicsek, Phys. Rev. Lett. 53 (1984) 2281. T. Vicsek, Phys. Rev. A 32 (1985) 3085. C. Tang, Phys. Rev. A 31 (1985) 1977. J. Kertesz and T. Vicsek, J. Phys. A 19 (1986) L257. J. Nittmann and H.E. Stanley, Nature 321 (1986) 663. J. Nittmann and H.E. Stanley, J. Phys. A 20 (1987) Ll185. H. Brune, C. Romainczyk, H. Roder and K. Kern, Nature 369 (1994) 469. H. Brune, C. Romainczyk, H. Roder and K. Kern, Phys. Rev. Lett. 74 (1995) 3217. R. Seshardri, G.N. Subanna, V. Vijayakrishnan, G.U. Kulkami, G. Ananthakrishna and C.N.R. Rao, J. Phys. Chem. 99 (1995) 5639.
Physica A 224 (1996) 412-421
Monomer aggregation model for DLA dendrite crossover G. Ananthakrishna, Silvester J. Noronha Materials Research Centre, Indian Institute of Science, Bangalore 560 012, India
Abstract We consider Monte Carlo simulation of monomer aggregation into clusters with a view to simulate both DLA and dendrites. We introduce a new method of suppressing fluctuations by allowing two types of relaxations. First is a local on-site relaxation of monomers and the second, diffusive relaxation of the monomers along the perimeter of the cluster. The former is characterized by two energies and the latter by two activation energies corresponding to diffusion along edges and diffusion motion which is transverse locally. The model produces both DLA- and dendritelike patterns for different choices of parameters. We find two distinct configurations of dendrites corresponding to two distinct anisotropic directions.
I. Introduction Patterns formed under non-equilibrium growth conditions have been an object of attention for a long time [ 1]. These include a rich variety of dendritic patterns which appear to be very regular, in addition to the randomly ramified structures [2,3]. Such patterns have been observed in physical, chemical and biological systems [3,4]. Typically, the growth occurs by transport of heat or material by diffusion. Therefore in both cases, solution of the Laplacian plays an important role. The main aim of theoretical models is to understand and reproduce these patterns. There are essentially two distinct approaches to these problems. The first one is basically a continuum deterministic approach wherein the essential physics is described by a heat conduction equation, the interfacial moving boundary condition and inclusion of surface tension effects [ 1,6]. Different types of patterns result basically from the competition of the enhanced growth due to non-local diffusion field of that part of the crystal which bulges into the melt and the surface tension effects which tend to flatten the interface. Numerical solution Elsevier Science B.V. SSD10378-4371 (95)00359-2
G. Ananthakrishna, S.J. Noronha/Physica A 224 (1996) 412-421
413
of these equations supplemented with an equation for the environmental factors already gives patterns that are very close to real snowflakes [5]. The anisotropic direction that appears to manifest is either a consequence of the surface tension anisotropy or that of the underlying lattice on which the Laplacian is solved. In the last few years there is an improved understanding of the various morphologies arising given the anisotropy and the undercooling [6]. In contrast, the usual DLA model which proceeds by adding monomers to the cluster in an irreversible way produces ramified patterns [2,3] which do not posses any symmetry. Even though, the latter is known be related to the growth of regular dendrites [2], it is not related in any obvious way. The later method can be termed as 'discrete', since it attempts to build the cluster by irreversible aggregation of monomers at a time. While the first method is suitable for the description of dendritic crystal, the second produces ramified patterns. Both these approaches have given a good deal of understanding of the formation of the patterns. Fluctuations appear to be suppressed in the case of dendrites while in the case of the DLA, fluctuations dominate. Since these two approaches start from different ends, there have been attempts to synthesize an intermediate approach which starts with DLA approach but attempts to suppress the fluctuations [7-13]. One of the first methods was to include relaxation of the particle to lowest energy configuration in the neighbourhood of the point where the particle arrives. They include an algorithm to mimic the surface tension effects [8,9]. One method which has been very successful in producing the crossover behaviour from DLA to dendrites, involves suppressing the fluctuations by using a counter which keeps track of the number of random walkers visiting a site on the perimeter [10-13]. A monomer is added once the counter exceeds a certain value m*. The limit m* --+ 1 produces the DLA, while m* ~ c~ produces compact clusters. Suppressing fluctuations coupled with using the solution of the Laplacian as in the dielectric breakdown model appears to produce realistic snowflake structures [ 11-13]. Again, in these studies the anisotropy arises due to the lattice structure. In such models as above, the effect of changes in the experimentally controlled parameters such as temperature, extent of supersaturation or undercooling cannot be easily understood. It is the purpose of the present work to come up with a model wherein the theoretical parameters have a direct correspondence with the experimental parameters. In this brief study we include only temperature related parameters. Even within the limited scope of the present model, we find both DLA and dendritic type of clusters. Surprisingly, we find that the morphology of dense branched dendrites picks out two equivalent anisotropic directions, one of which is natural to the nucleus and the other to the line joining the corners.
2. The model Consider a monomer performing an off-lattice random walk in two dimensions. Each monomer is considered to have a finite size and is assumed for the sake of convenience to be of square shape with an edge length denoted by 8. (Other shapes can also be con-
414
G. Ananthakrishna, S..I. Noronha/Physica A 224 (1996) 412-421
(a)
(b) i .....
-.'. . . . . . . . . . .
' ...............
,~ . . . . . . . . .
J ...........
J
.........
i ....
J
i
i
! ' !l
Fig. 1. (a) and (b) refer to the two differentinterpretations of the on-site relaxation. Only (b) is relevant for the model for which the results are presented. However,no spatial relaxation is carried out in the simulation. sidered.) As in any DLA simulation, the monomers are released from the circumference of the circle whose radius is kept large. We do not include any orientational dynamics while the monomers are diffusing towards the aggregate, i.e., the edges of the monomers remain parallel to the initial directions of the released monomers. A single monomer located at the origin (with its center of mass coinciding with the origin) with its edges parallel to the x - y axes acts as the nucleus. In this model we allow the aggregation of the monomer such that the edges of the monomer match with the nucleus (or any of the free edges of a monomer which is part of the aggregate). We accomplish this in the following way. Consider a monomer reaching the neighbourhood of the nucleus within 26 (measured from the respective center of masses of the particles involved) for the first time. Since the diffusive motion of the monomer is an off-lattice random walk, the distance between the center of mass of the nucleus and the monomer will not be in general an integer multiple of the edge length. From conventional molecular dynamics calculations on any aggregation process, we know that particles move in a potential which is a sum of two body potentials depending on the distances between the particles. Only when they occupy the minimum potential energy configuration the clusters are considered stable. We believe that there is a similar situation in the present aggregation process also. Thus, we assume that once the monomer arrives for the first time within a distance 26 from the nucleus, it experiences a force of attraction which increases as a function of its distance, reaching a maximum when the distance between them is 6. We refer to this as on-site relaxation. This in principal involves both rotational relaxation as well as changes in the distance between the center of mass of the nucleus and the monomer such that the edges eventually become parallel and match. This is shown in Figs. l a,b. Preliminary simulations which were initially carried out showed that these simulations were very time consuming. Therefore, we have simplified this and several other aspects o f the above model as stated below: a) We allow the aggregation to proceed by matching of the edges of the monomers with the monomers belonging to the perimeter sites of the aggregate. b) The first simplification effected is that the monomers are released from the circumference of the circle with the edges of the monomers always parallel to the nucleus. Since we do not have any orientational dynamics in the diffusion of the monomers, they always arrive in the neighbourhood (see below) of the nucleus with their edges parallel to the edges of the nucleus. Even as the aggregation proceeds,
G. Ananthakrishna, S.J. Noronha/Physica A 224 (1996) 412-421
415
the monomers arrive with their edges parallel to the monomers at the perimeter sites of the aggregate. c) The perimeter sites of the aggregate are defined as those sites which are not interior sites of the aggregate and therefore have less than three monomers (belonging to the aggregate) as their neighbours, (Note that in the aggregate, the centers of mass of the neighbours are all at distance 8.) Once the monomer arrives at a distance within 28 from any perimeter site (it could be more than one perimeter site) we consider that it is subject to a force of attraction as stated above. However, we do not explicitly carry out the movement of the monomer, but we keep track of the changes in the energy only, as a function of time. This is adequate for the present Monte-Carlo calculation since we need to effect the addition of the monomer at the site of the aggregate only when the monomer has a sticking probability equal to unity. The time-dependent binding energy is taken to be bn(t) = bn(0) + d[1 - e x p ( - t / ~ ' ) ].
(i)
Here, bn(t) is the binding energy of the monomer at the nth site at time t, bn(O) is the binding energy at time t = 0 (corresponding to a distance 8 from the perimeter site), ~- is the time scale of relaxation and A + bn(O) refers to the limiting binding energy of the bond established by the monomer with the cluster on sticking. (See Fig. lb.) d) The number of bonds established by the monomer is taken to be equal to the number of distances from the center of mass of the monomer to all possible perimeter site monomers of the aggregate which are less than 28. We further assume that all of them have the same value as long as these distances are less than 28. Corresponding to a fiat interface, there is only one bond (Fig. 2a). If there is a step as in Fig. 2b, the monomer can have two bonds. We take the particle to have three bonds corresponding to a well (Fig. 2c). e) The monomer can either stay at the site and relax on-site if it does not hop to its neighbouring sites, or it can hop to the neighbouring sites. Note that t refers to the time for relaxation at that site. If the particle moves to a neighbouring site, the time for the on-site relaxation starts at zero for that site. When the particle arrives within 28 length of the aggregate, it can also diffuse along the perimeter sites with a probability of migration proportional to e x p ( - E m / k T ) . We introduce two distinct energies of migration, one corresponding to migration on a fiat interface, E,(e) (Fig. 2a) and another to migration around a sharp corner, E(mr) (Figs. 2d,e). These two physically correspond to diffusive motion parallel and perpendicular to any branch of the aggregate. Monte Carlo simulation of the aggregation is carded out by the conventional energy minimization. The probability of hopping from the nth to the (n 4- 1 )th site is given by Pn,n+l O~ e -(E'+ae','±t)/kT",
(2)
416
G. Ananthakrishna, S.J. Noronha / Physica A 224 (1996) 412-421
(b)
(a) i.......... i ~......... d
1...........l...........!.........l................. t..........l.............'i
col
(0)
i.........J
...........J
tf!ll'i ~
i
,
i
1
i
[
.....
I
(f)
(e)
...........
I ~!
r
J ~
,iii[
Fig. 2. Distinct types of configurations of the monomer with the aggregate.
where E m = E,~e) or E~r) depending on the position of the particle. AEn,n+l = b , ( 0 ) + A [ l - e x p ( - t / r ) ] - b , ± l (0) is the change in the energy between the initial and the final states. Fig. 2a corresponds to migration along flat edges and Figs. 2d or 2e correspond to diffusive motion around sharp corners which locally is transverse to the motion along the edge. The probability of staying at the site ps = 1 - P,,,-1 - Pn.n+l. Due to the fact that there is an on-site time-dependent relaxation, we consider the particle to stick to a perimeter site if the on-site staying probability is more than a certain fixed number p*. This parameter has been essentially introduced to minimize the computational time. In our calculations this number is chosen to be between 0.8 to 0.95. Figs. 2a to 2e show typical configurations that are possible. From Eq. (2) it is clear that in the case of Fig. 2a, the monomer can move to the left or to the right with equal probability as it arrives at a distance 6 from the cluster (i.e. at t = 0). If it does not hop to the neighbouring site, then it is clear that the probability of staying at the site increases as a function of time. Further, at any later time, if the particle hops to a neighbouring site, we assume that the monomer hops to the least relaxed state. Consider the situation shown in Fig. 2f. From Eq. (2), the monomer has a higher probability to
G. Ananthakrishna, S.J. Noronha/ Physica A 224 (1996) 412-421
Fig. 3. DLA structure produced for
E~)IkT = E(mr)/kT = 4, bn(O)/kT
= 1.0,
A/kT
417
= 2.0, p * = 0.8 and
7"=1.
move to the left than to the right. In addition, having arrived there (Fig. 2b), it has a greater probability to eventually stick there. Consider the situation shown in Figs. 2d and 2e. The energy of migration for this situation is E~r~. The possible values E~mr~ can take depend on the actual physical situation. For our purposes we consider all possibilities, E(me) > E (r) , E(me) = E~~ and E (e) < E (e). Note that while E~m~ governs the diffusive motion of the particle along the branch of the aggregate, E~mr~ refers to the local transverse motion of the aggregate. Note, also Fig. 2c corresponds to the lowest energy configuration.
3. Results and discussion
All energies we use are in scaled units of kT. It is clear that when both the migration energies E ~e~ and E~,~ are large, we should expect that the particle has very little tendency to diffuse along the perimeter of the aggregate. Therefore, we should see a DLA-like structure when a binding energy, b0 and A are reasonable. Fig. 3 shows a typical DLA-like structure. The effects of A and bn (0) are interesting. If A is small, the on-site relaxation is not significant. If now the energy of migration E~e~ is decreased favouring diffusion along the fiat portion of the perimeter of the aggregate, thick branches can arise, if the bond energy bn(O) is reasonable. This is because the monomers have a tendency to aggregate at steps (as in Fig. 2c). A typical such densely branched morphology is shown in Fig. 4. In this case, the value of E~r~ chosen is unfavourable for the monomers to diffuse round sharp corners. This leads to no preferential growth direction (Fig. 4). If we further allow a decrease in E~e~, it favours rapid diffusion along the fiat parts of the aggregate, if in addition we choose E~r~ to be large, the monomer has
418
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Fig. 4. Dense branched nearly symmetric pattern produced for E~me)/kT= 0.5, E~mr)/kT= 2, bn(O)/kT = 1.0, zl/kT = 0.05, p* = 0.8 and ~"= 1.
Fig. 5. Dendrite type morphology produced with x-y anisotropic direction for E(e)/kT = 0.01, E~r)/kT = 3, bn(O)/kt = 1.0, zl/kT = 0.05, p* = 0.8 and T = I. very little probability o f m o v e m e n t round the corners. This aids the growth of the cluster in the x - y directions, thereby p i c k i n g out the anisotropic growth directions seen in Fig. 5. This direction essentially corresponds to the natural anisotropic direction o f the nucleus at the origin. C o n s i d e r n o w a situation wherein both migration energies, corresponding to diffusion along and perpendicular to the branch are chosen to be o f the same order o f magnitude. A typical such figure is shown in Fig. 6. The aggregate has a dense branched m o r p h o l o g y typical to many dendritic crystals with an anisotropic direction at 45 ° to the orignal nucleus. This anisotropic directionality is surprising considering
the fact that the nucleus does not posses this symmetry. Clearly, this direction has been
G. Ananthakrishna, S.J. Noronha/ Physica A 224 (1996) 412-421
419
Fig. 6. Thick branched dendrite type morphology produced at 45° to the nucleus along the x-y. E~e)/kT = E~)/kT = 1, bn(O)/kT = 1.0, A/kT = 2.0, p* = 0.8 and r = 1.
Fig. 7. Dendritic structure produced for E~e)/kT = 0.5, E~mr)/kT= 3, bn(O)/kT = 0.5, zl/kT = 0.5, p* = 0.8 and r = 8.
picked out due to the competition between two directions of diffusion. It must be noted that in this case, the thickness o f the branches are much more than in the case o f Fig. 5. The effect o f the relaxation time r also has an important effect on the morphology of the aggregate. For the set o f parameters when the growth is preferentially in the x - y directions, increasing 7- gives rise to several branches with the aggregate having nearly a square shape. This is shown in Fig. 7. Lastly, the effect of increasing the bn(0) is to make the branches thicker as shown in Fig. 8. As mentioned earlier, the model in its present form does not include other effects such as the flux o f the incoming particles etc. Refinements of the model to various physical situations are under way. Here, we would like to emphasize that our attempt is not so much to produce realistic snowflake-like patterns, but to examine how the crossover arises as a function o f various physical parameters. Even so, the preliminary results are sufficiently encouraging that we expect that this model with appropriate modifications should be able to describe some physical situations. For instance, we expect that once the
420
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Fig. 8. Thick branched dendrite structure produced for E(me)/kT = 0.5, E~r)/kT = 3.0, bn(O)/kT = 2.0, A/kT = 0.5, p* = 0.8 and ~"= 1. flux rate and circular geometry o f the particles are included, the model should be able to describe the recent results on the growth o f A g clusters in molecular vapour deposition on Pt (111) or A g (111) substrate [14,15]. Another instance we can visualize is a situation wherein the monomers are diffusing in a solution and aggregating to form different types o f patterns such as the D L A patterns and dendrites. Indeed, such a situation has been recently seen in the aggregation o f gold monomers [ 16]. We are in the process o f refining the model to the actual physical situation. In conclusion, in this model, we have devised an alternate method o f suppressing fluctuations which is physically closer to real situations. This has been accomplished by allowing for the diffusive relaxation coupled with energy minimization. This helps to sample a large number o f energy states. The D L A limit in this model arises when the migration energies are high and hence does not allow for suppression o f fluctuations. A cross over to dendritic type structure having two distinct anisotropic directions is shown to arise as a consequence o f competition o f two different energies o f migration. When diffusion in the transverse direction is prevented (E{me) << E{mr}), the dendrite picks out the natural anisotropic direction of the nucleus. In contrast, when both transverse and parallel directions o f diffusion compete, a new anisotropic direction which is not natural to the monomer is picked up. Further, the model has parameters which are experimentally accessible and have easy physical interpretation.
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