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Computer simulation of growing fractal nanodendrites by using of the multi-directed cellular automatic device
Materials Science and Engineering C 27 (2007) 1270 – 1272 www.elsevier.com/locate/msec
Computer simulation of growing fractal nanodendrites by using of the multi-directed cellular automatic device S.A. Beznosyuk ⁎, Y.V. Lerh, T.M. Zhukovsky, M.S. Zhukovsky Department of Physical Chemistry, Altai State University, 61, Lenin Av., 656049, Barnaul, Russian Federation Received 28 April 2006; received in revised form 15 August 2006; accepted 20 August 2006 Available online 1 November 2006
Abstract Computer simulation of self-assembly growth of fractal nanodendrites is suggested by using a new model of multi-directed cellular automatic device. The novelty is that in a framework of the multiparticle diffusion limited aggregation (DLA) model, atoms stick to the growing cluster according to specific kinematics multi-directional contact bonds. The aggregation is limited by initial concentration of particles confined inside definite nano-sized enclosures. Self-organizing process results in the formation of fractal dendrite nanostructures. From results of computer experiments, some equations for cluster fractal dimension D, depending on the initial concentration C of diffusing particles in 2d and 3d space enclosures, are deduced. Correlations between cluster fractal dimension D and number of aggregated particles N are found also. © 2006 Elsevier B.V. All rights reserved. Keywords: Computer modeling of mechanism of diffusion-limited aggregation; Multi-directional cellular automatic devices; Self-assembly of fractal nanodendrites
1. Introduction Future nanotechnologies will be based on fast-growing mainframes and assembly of self-organized multistructural functional materials [1]. One of the base nanotechnological stocks is an atomic nanodendrite characterized by fractal dimension [2]. A model for random aggregate nanodendrite is usually studied by computer simulation in a framework of DLA, which is an extremely influential model for such kind of investigations [3]. In particular, this model is applicable to a metalparticle aggregation process. There are some models of DLA, in which it was possible to set concentration of diffusing particles [4]. An unsteady effect of the diffusion on DLA was studied using the multiparticle DLA model on the deposition plate in two dimensions [5]. The multiparticle model can take into account the propagation of diffusing particles from the upper boundary toward the deposition plate. The morphological evolution was investigated by means of computer simulations. It was shown that with increasing concentration, the morphology of the deposit becomes different from that grown in a steady state. A parallel algorithm for DLA was described and analyzed [6]. The ⁎ Corresponding author. E-mail address: [email protected] (S.A. Beznosyuk). 0928-4931/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msec.2006.08.013
dynamic exponent z of the algorithm was defined with respect to the probabilistic parallel random-access machine model of parallel computation according to T ∼ Lz, where L is the cluster size, T is the running time, and the algorithm uses a number of processors polynomial in L. It was argued that z = D − D2/2, where D is the fractal dimension and D2 is the second generalized dimension. Simulations of DLA were carried out to measure D2 and to test scaling assumptions employed in the complexity analysis of the parallel algorithm. The given research is the further development of previous works. Distinctive feature of research is that it is directed on studying formation of nonequilibrium dendrite nanostructures in special conditions. These conditions simulate formation of self-organizing cluster of a set number of atoms in strictly certain point of growth inside strictly definite (square or cubic) enclosures. In addition, such aggregation of atoms is supervised not only by propagation of diffusing particles, but also by special rules of their sticking to the growing unit. In accordance with a new model of multi-directed cellular automatic devices, the rule of sticking depends on various discrete directions of atomic collision. They are defined with specific kinematics restrictions on contacts between atoms having “square” form in 2d-space or “cubic” form in 3d-space. Novelty is also that within the framework of these approaches we obtain regularity
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In circumstances, there is a probability of recalculation. Then in the cell in which the particle moves is occupied, there will be no free direction of movement. If the particle does not have free direction, it remains in its place. These conditions eliminate stratification of points against each other. After scoring a hit in the nearest vicinity of growing nanodendrite, each wandering particle becomes a part of its structure with a probability defined with a direction of its attack. The last conditions specify the graph of specific kinematic rules of atomic stick to the growing dendrite. In the offered computer model, a particle has three possible ways of linkage with growing cluster (top, edge, and side). A fractal dimension of the final dendrite cluster is calculated by using the geometrical method [2]. This method is most available in case of grids. The fractal dimension D is defined from double logarithmic coordinates dependence, which is given by the formula: log2 N = log2 b − D·log2 L, where L is the length of the super-cell of an imposed super-lattice on the grid, N is the number of the borrowed super-cells, and b is a model constant. 3. Results and discussion
Fig. 1. Final nanodendrites in 2d and 3d spaces: (a) the size of the area is 150 × 150, the center of aggregation–(75, 75), C = 5%, N = 1001, D = 1.49; (b) the size of the area is 50 × 50 × 50, the center of aggregation–(25, 25, 25), C = 1%, N = 1001, D = 1.82.
dependence of final nanodendrite fractal dimensions from initial distributions of diffusing particles. 2. Computer Modeling In initial time, the image set of initial distributions of diffusing particles is specified on a definite lattice. The germ is located in a central point of the lattice. The fractal dendrite grows either on a substrate of the finite size (Fig. 1a) or inside three-dimensional subspace (Fig. 1b) as a result of propagation of diffusing particles, and their multi-directed sticking to the growing unit. Concentration of particles remains constant, and particles never leave borders of definite area. In case of propagation of diffusing “square”-formed particle on the deposition plate at the subsequent step, it occupies the position in one of eight adjacent cells. In the three-dimensional version of this model, the “cubic”-formed particle can move in 26 directions. Particles can move only to those cells, which are not occupied both in the given time step, and in the subsequent one.
For concentration of 5%, 10% and 15%, the two-dimensional variant of model is carried out on square grids with the size 150 × 150 of deposition plate. There is a single nucleating center of aggregation in a point (75, 75). In cases of other concentrations (20–100%) modeling is fulfilled on square lattices with the size 100 × 100, which have a single nucleating center of aggregation at point (50, 50). In computer experiments, for each concentration C, three dendrites are grown. Fractal dimensions of the synthesized aggregates are subjected to standard statistical mathematical processing. It is found that the equation of dependence between the mean fractal dimension D and initial concentration of particles C has a form: D = 1.476 + 0.00493C (see Fig. 2). Coefficient of correlation for this equation, R, is equal to 0.999. Experimental errors in the mean of D are not more than 0.04. Received dependence shows that Dmin = 1.48 and Dmax = 1.97. For final dendrite fractal with classical fractal dimension on a plane (D = 1.71), the initial concentration of particles is equal to 47.5%. With increasing initial concentration of particles above 65.7% fractal dimension of final cluster becomes more than 1.8,
Fig. 2. Dependence of fractal dimension from initial concentration in 2d space.
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of aggregated particles N = 103–105 fractal dimension lies within the interval from D = 1.87 up to D = 2.11. 4. Conclusion
Fig. 3. (a) Dependence of fractal dimension from initial concentration of particles in 3d space. (b) Dependence of fractal dimension from the number of aggregated particles in 3d space.
and in the limit of C = 100%, the structure of the deposit agrees with the two-dimensional structures of sheet. The three-dimensional variant of the model is carried out on finite cubic grids in the size 50 × 50 × 50 of an enclosure. There is a single nucleating center of aggregation in a point (25, 25, 25). Modeling is only realized for initial concentration of 5%, 10%, 15%, 20% and 25%. The equation of dependence between the fractal dimension and the initial concentration is found, as follows: D = 1.80 + 0.024C with coefficient of correlation equal to 0.996. The plot of this dependence is shown in Fig. 3(a). With incising initial concentration below 8.3%, the fractal dimension of the final dendrites becomes smaller than one for the sheet: D = 1.8–2.0. We take also under consideration the process of diffusion of particles inside interspaces between dendrite branches. For this purpose, virtual nanostructures are synthesized in an area of 100 × 100 × 100 cells with the start center of aggregation in a point (50, 50, 50). We choose numbers N of the aggregated particles as follows: 1036, 2514, 5031, 7585, 10,090, 12,699, and 15,081. Fractal dimension D depends on the number N of aggregated particles according to the nonlinear equation: D = 1.76 + 1.27 × 10 − 4 N − 1.41 × 10 − 8 N 2 + 4.94 × 10 − 13 N 3 . Nonlinear dependence is plotted in Fig. 3(b). The coefficient of correlation for this equation R is equal to 0.978. For the number
In this work, a new multi-directed cellular automatic device model describing growth of fractal nanodendrite is constructed. The novelty of the model consists of as follows. Firstly, atomic aggregation occurs on the fixed set center inside strictly limited nano-sized two- and three-dimensional areas of space. Secondly, the number aggregating atoms is strictly limited by their initial concentration. Thirdly, aggregating process is supervised not only by stochastic propagation of diffusing atoms, but also by specific kinematics rules of atomic stick to the growing dendrite. There are three interesting results of our computer experiment in the case of 3-dimentional finite enclosures. Firstly, dendrite branching is lower and its branches are thinner in the case of small concentration of particles. Increase of concentration leads to increase of branch's thickness. In a limit of high initial concentrations, the nanostructure of dendrite becomes to be similar to an amorphous lump. Secondly, a self-organizing nanodendrite possesses an unusual property. Their final fractal dimension increases with an increase in initial number of the aggregated particles under the nonlinear law unlike linear decreasing of potential energy of interatomic bonds. It specifies that binding energy of the nanodendrite cluster is the completely additive measure of its unsteadiness. Whereas it is plausible that fractal dimension sets nanodendrite cluster structure complexity index. Thirdly, from the set above the results, it is possible to draw a conclusion that the evolution of the cellular automatic device from the set initial DLA model atomic configurations can be considered as the consecutive calculation processing the information, which is contained with a configuration of dendrite. In the given work, it is shown, that in cases when the model of cellular automatic devices are supplied by “correct” multidirected kinematics stick restrictions, it is really to obtain nonlinear dependences between the final cluster fractal dimension D and the number of aggregated particles N. On the other hand, the restrictions of the given cellular automatic device are ideally suitable for the realization on the computer possessing a high degree of parallelism, locality and uniform interrelations. The last is very important to get at the root of the nanostructural mechanisms conditioning adaptive behavior of smart materials. References [1] S.A. Beznosyuk, Mater. Sci. Eng., C, Biomim. Mater., Sens. Syst. 19 (1) (2002) 369. [2] Feder E. Fraktals.- M.: MIR, 1991. 260 p. [3] T.A. Witten, L.M. Sander, Phys. Rev. Lett. 47 (1981) 1400. [4] T. Toffoli, Machine of cellular automatic devices/T. Toffoli, N. Margolus.M. MIR, 1991. 278. [5] T. Nagatani, J. Phys. Soc. Jpn. 61 (5) (1992) 1437. [6] K. Moriarty, J. Machta, R. Greenlaw, Phys. Rev. 55 (1997) 6211.
Computer simulation of growing fractal nanodendrites by using of the multi-directed cellular automatic device S.A. Beznosyuk ⁎, Y.V. Lerh, T.M. Zhukovsky, M.S. Zhukovsky Department of Physical Chemistry, Altai State University, 61, Lenin Av., 656049, Barnaul, Russian Federation Received 28 April 2006; received in revised form 15 August 2006; accepted 20 August 2006 Available online 1 November 2006
Abstract Computer simulation of self-assembly growth of fractal nanodendrites is suggested by using a new model of multi-directed cellular automatic device. The novelty is that in a framework of the multiparticle diffusion limited aggregation (DLA) model, atoms stick to the growing cluster according to specific kinematics multi-directional contact bonds. The aggregation is limited by initial concentration of particles confined inside definite nano-sized enclosures. Self-organizing process results in the formation of fractal dendrite nanostructures. From results of computer experiments, some equations for cluster fractal dimension D, depending on the initial concentration C of diffusing particles in 2d and 3d space enclosures, are deduced. Correlations between cluster fractal dimension D and number of aggregated particles N are found also. © 2006 Elsevier B.V. All rights reserved. Keywords: Computer modeling of mechanism of diffusion-limited aggregation; Multi-directional cellular automatic devices; Self-assembly of fractal nanodendrites
1. Introduction Future nanotechnologies will be based on fast-growing mainframes and assembly of self-organized multistructural functional materials [1]. One of the base nanotechnological stocks is an atomic nanodendrite characterized by fractal dimension [2]. A model for random aggregate nanodendrite is usually studied by computer simulation in a framework of DLA, which is an extremely influential model for such kind of investigations [3]. In particular, this model is applicable to a metalparticle aggregation process. There are some models of DLA, in which it was possible to set concentration of diffusing particles [4]. An unsteady effect of the diffusion on DLA was studied using the multiparticle DLA model on the deposition plate in two dimensions [5]. The multiparticle model can take into account the propagation of diffusing particles from the upper boundary toward the deposition plate. The morphological evolution was investigated by means of computer simulations. It was shown that with increasing concentration, the morphology of the deposit becomes different from that grown in a steady state. A parallel algorithm for DLA was described and analyzed [6]. The ⁎ Corresponding author. E-mail address: [email protected] (S.A. Beznosyuk). 0928-4931/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msec.2006.08.013
dynamic exponent z of the algorithm was defined with respect to the probabilistic parallel random-access machine model of parallel computation according to T ∼ Lz, where L is the cluster size, T is the running time, and the algorithm uses a number of processors polynomial in L. It was argued that z = D − D2/2, where D is the fractal dimension and D2 is the second generalized dimension. Simulations of DLA were carried out to measure D2 and to test scaling assumptions employed in the complexity analysis of the parallel algorithm. The given research is the further development of previous works. Distinctive feature of research is that it is directed on studying formation of nonequilibrium dendrite nanostructures in special conditions. These conditions simulate formation of self-organizing cluster of a set number of atoms in strictly certain point of growth inside strictly definite (square or cubic) enclosures. In addition, such aggregation of atoms is supervised not only by propagation of diffusing particles, but also by special rules of their sticking to the growing unit. In accordance with a new model of multi-directed cellular automatic devices, the rule of sticking depends on various discrete directions of atomic collision. They are defined with specific kinematics restrictions on contacts between atoms having “square” form in 2d-space or “cubic” form in 3d-space. Novelty is also that within the framework of these approaches we obtain regularity
S.A. Beznosyuk et al. / Materials Science and Engineering C 27 (2007) 1270–1272
1271
In circumstances, there is a probability of recalculation. Then in the cell in which the particle moves is occupied, there will be no free direction of movement. If the particle does not have free direction, it remains in its place. These conditions eliminate stratification of points against each other. After scoring a hit in the nearest vicinity of growing nanodendrite, each wandering particle becomes a part of its structure with a probability defined with a direction of its attack. The last conditions specify the graph of specific kinematic rules of atomic stick to the growing dendrite. In the offered computer model, a particle has three possible ways of linkage with growing cluster (top, edge, and side). A fractal dimension of the final dendrite cluster is calculated by using the geometrical method [2]. This method is most available in case of grids. The fractal dimension D is defined from double logarithmic coordinates dependence, which is given by the formula: log2 N = log2 b − D·log2 L, where L is the length of the super-cell of an imposed super-lattice on the grid, N is the number of the borrowed super-cells, and b is a model constant. 3. Results and discussion
Fig. 1. Final nanodendrites in 2d and 3d spaces: (a) the size of the area is 150 × 150, the center of aggregation–(75, 75), C = 5%, N = 1001, D = 1.49; (b) the size of the area is 50 × 50 × 50, the center of aggregation–(25, 25, 25), C = 1%, N = 1001, D = 1.82.
dependence of final nanodendrite fractal dimensions from initial distributions of diffusing particles. 2. Computer Modeling In initial time, the image set of initial distributions of diffusing particles is specified on a definite lattice. The germ is located in a central point of the lattice. The fractal dendrite grows either on a substrate of the finite size (Fig. 1a) or inside three-dimensional subspace (Fig. 1b) as a result of propagation of diffusing particles, and their multi-directed sticking to the growing unit. Concentration of particles remains constant, and particles never leave borders of definite area. In case of propagation of diffusing “square”-formed particle on the deposition plate at the subsequent step, it occupies the position in one of eight adjacent cells. In the three-dimensional version of this model, the “cubic”-formed particle can move in 26 directions. Particles can move only to those cells, which are not occupied both in the given time step, and in the subsequent one.
For concentration of 5%, 10% and 15%, the two-dimensional variant of model is carried out on square grids with the size 150 × 150 of deposition plate. There is a single nucleating center of aggregation in a point (75, 75). In cases of other concentrations (20–100%) modeling is fulfilled on square lattices with the size 100 × 100, which have a single nucleating center of aggregation at point (50, 50). In computer experiments, for each concentration C, three dendrites are grown. Fractal dimensions of the synthesized aggregates are subjected to standard statistical mathematical processing. It is found that the equation of dependence between the mean fractal dimension D and initial concentration of particles C has a form: D = 1.476 + 0.00493C (see Fig. 2). Coefficient of correlation for this equation, R, is equal to 0.999. Experimental errors in the mean of D are not more than 0.04. Received dependence shows that Dmin = 1.48 and Dmax = 1.97. For final dendrite fractal with classical fractal dimension on a plane (D = 1.71), the initial concentration of particles is equal to 47.5%. With increasing initial concentration of particles above 65.7% fractal dimension of final cluster becomes more than 1.8,
Fig. 2. Dependence of fractal dimension from initial concentration in 2d space.
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of aggregated particles N = 103–105 fractal dimension lies within the interval from D = 1.87 up to D = 2.11. 4. Conclusion
Fig. 3. (a) Dependence of fractal dimension from initial concentration of particles in 3d space. (b) Dependence of fractal dimension from the number of aggregated particles in 3d space.
and in the limit of C = 100%, the structure of the deposit agrees with the two-dimensional structures of sheet. The three-dimensional variant of the model is carried out on finite cubic grids in the size 50 × 50 × 50 of an enclosure. There is a single nucleating center of aggregation in a point (25, 25, 25). Modeling is only realized for initial concentration of 5%, 10%, 15%, 20% and 25%. The equation of dependence between the fractal dimension and the initial concentration is found, as follows: D = 1.80 + 0.024C with coefficient of correlation equal to 0.996. The plot of this dependence is shown in Fig. 3(a). With incising initial concentration below 8.3%, the fractal dimension of the final dendrites becomes smaller than one for the sheet: D = 1.8–2.0. We take also under consideration the process of diffusion of particles inside interspaces between dendrite branches. For this purpose, virtual nanostructures are synthesized in an area of 100 × 100 × 100 cells with the start center of aggregation in a point (50, 50, 50). We choose numbers N of the aggregated particles as follows: 1036, 2514, 5031, 7585, 10,090, 12,699, and 15,081. Fractal dimension D depends on the number N of aggregated particles according to the nonlinear equation: D = 1.76 + 1.27 × 10 − 4 N − 1.41 × 10 − 8 N 2 + 4.94 × 10 − 13 N 3 . Nonlinear dependence is plotted in Fig. 3(b). The coefficient of correlation for this equation R is equal to 0.978. For the number
In this work, a new multi-directed cellular automatic device model describing growth of fractal nanodendrite is constructed. The novelty of the model consists of as follows. Firstly, atomic aggregation occurs on the fixed set center inside strictly limited nano-sized two- and three-dimensional areas of space. Secondly, the number aggregating atoms is strictly limited by their initial concentration. Thirdly, aggregating process is supervised not only by stochastic propagation of diffusing atoms, but also by specific kinematics rules of atomic stick to the growing dendrite. There are three interesting results of our computer experiment in the case of 3-dimentional finite enclosures. Firstly, dendrite branching is lower and its branches are thinner in the case of small concentration of particles. Increase of concentration leads to increase of branch's thickness. In a limit of high initial concentrations, the nanostructure of dendrite becomes to be similar to an amorphous lump. Secondly, a self-organizing nanodendrite possesses an unusual property. Their final fractal dimension increases with an increase in initial number of the aggregated particles under the nonlinear law unlike linear decreasing of potential energy of interatomic bonds. It specifies that binding energy of the nanodendrite cluster is the completely additive measure of its unsteadiness. Whereas it is plausible that fractal dimension sets nanodendrite cluster structure complexity index. Thirdly, from the set above the results, it is possible to draw a conclusion that the evolution of the cellular automatic device from the set initial DLA model atomic configurations can be considered as the consecutive calculation processing the information, which is contained with a configuration of dendrite. In the given work, it is shown, that in cases when the model of cellular automatic devices are supplied by “correct” multidirected kinematics stick restrictions, it is really to obtain nonlinear dependences between the final cluster fractal dimension D and the number of aggregated particles N. On the other hand, the restrictions of the given cellular automatic device are ideally suitable for the realization on the computer possessing a high degree of parallelism, locality and uniform interrelations. The last is very important to get at the root of the nanostructural mechanisms conditioning adaptive behavior of smart materials. References [1] S.A. Beznosyuk, Mater. Sci. Eng., C, Biomim. Mater., Sens. Syst. 19 (1) (2002) 369. [2] Feder E. Fraktals.- M.: MIR, 1991. 260 p. [3] T.A. Witten, L.M. Sander, Phys. Rev. Lett. 47 (1981) 1400. [4] T. Toffoli, Machine of cellular automatic devices/T. Toffoli, N. Margolus.M. MIR, 1991. 278. [5] T. Nagatani, J. Phys. Soc. Jpn. 61 (5) (1992) 1437. [6] K. Moriarty, J. Machta, R. Greenlaw, Phys. Rev. 55 (1997) 6211.