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The ‘random’ square–triangle tiling: simulation of growth
Materials Science and Engineering 294–296 (2000) 418–420
The ‘random’ square–triangle tiling: simulation of growth Boaz Rubinstein∗ , Shelomo I Ben-Abraham Department of Physics, Ben-Gurion University of the Negev, PO Box 653, IL-84105 Beer-Sheba, Israel Received 3 September 1999; accepted 6 October 1999
Abstract Some alloy systems, such as Ni–Cr, V–Ni–Si and Ta–Te, have quasicrystalline phases with 12-fold symmetry. These structures may be described in terms of dodecagonal tilings by equilateral triangles and squares. The formation of quasicrystals still poses a problem, since local information is insufficient for the construction of a perfect quasiperiodic structure. The growth of real quasicrystals may be due to several mechanisms. We have simulated the growth of a quasicrystal from a melt, consisting of squares and equilateral triangles of equal edge length. We are interested in the abundancies of the vertex configurations formed, both regular and defective. Unrestricted random growth tends to result in segregation of triangles from squares. Favoring triangles to attract squares and vice versa brings about nearly perfect patterns with nearly perfect vertex abundancies, as well as realistic defect concentrations. We have also calculated the exact vertex frequencies of the ideal square–triangle tiling by relying on inflation symmetry. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Dodecagonal tilings; Vertex; Inflation symmetry
1. Introduction Strictly local information is insufficient for the construction of a perfect generic quasiperiodic structure. Hence it is of continuing interest to study the growth mechanisms of real quasicrystals. One plausible scenario is building up the quasicrystal from some relatively stable structural units pre-existing in the melt through some suitable local interactions. The resulting structure can be abstractly represented by a tiling with some degree of randomness. Its formation can be viewed as a successive agglomeration of building blocks under some more or less stringent constraints. One approach is joining tiles from a given set face to face. An alternative is covering space by partially overlapping clusters [1–2]. According to a complementary model the crystal solidifies from the melt by adding single atoms, as opposed to clusters. The nucleus is allowed to freeze and remelt until equilibrium is reached (cf. [3–7]). In reality, different scenarios may apply to different systems. However, it is plausible that within the same system several mechanisms may compete, or more likely, act in synergy. We try to study these topics by simulating various growth mechanisms. Here we present a simulation of constrained random growth of the dodecagonal square–triangle tiling. The latter is an adequate abstract representation of the do-
∗ Corresponding author. Tel.: +972-7-6472419; fax: +972-7-6472903. E-mail address: [email protected] (B. Rubinstein).
decagonal phases observed in such alloy systems as Ni–Cr [8–10,15], V–Ni and V–Ni–Si [11], Ta–Te [12].
2. Procedure and results The rationale of our simulation is to study the growth of a dodecagonal quasicrystal from a melt. In order to represent reality, the growth should be visualized as an aggregation of pre-existing complex building blocks (clusters) rather than of individual atoms. That, of course, neither explicitly shows up in the tiling model nor in the simulation which is based upon it. The simulation proceeds according to the following simple algorithm: chose an edge at random on the free surface of the growing nucleus and attach a tile to it according to some predetermined rules. If the tile does not overlap other tiles, and is allowed by the algorithm, let it stay, else erase it. Then return to the first step. For technical reasons the whole structure is contained within a rectangular box of area 400 square units, the unit being an edge length. To mimic the chemistry we have tried a variety of restrictions. Among those were forbidding crystallographic vertices, and using “house-shaped” supertiles as building blocks. These consisted of a triangle attached onto a square. We have also varied the “chemical composition” of the melt, that is the ratio of triangles to squares in the melt. Unrestricted growth tended to produce phase separation between domains containing only triangles from domains
0921-5093/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 1 1 4 2 - 4
B. Rubinstein, S.I. Ben-Abraham / Materials Science and Engineering 294–296 (2000) 418–420
419
Fig. 3. Stampfli’s dodecagonal square–triangle tiling. Fig. 1. Phase √ separation in solidification from a melt, of square-to-triangle ratio of 3/4.
of squares, both, of course, periodic by themselves (Fig. 1). Most variations of the growth rules were not particularly effective. However, the simple device of forcing triangles to join squares and vice versa brought about a dramatic improvement. Under this constraint, if the chosen edge belongs to a triangle then attach a square, and vice versa. For example, the vertex 33222 (the digits code for the angles around
Fig. 2. Tiling produced by simulation of growth under the squareto-triangle constraint.
the vertex in units of 2π/12=30◦ ) cannot be built alone, and must be embedded in a larger structure; hence it indeed occurs rather sparsely. The resulting tilings showed a strong preference for local patterns occurring in nature (Fig. 2). A very similar observation was made by Kuo et al. [13], who arrived at their constraint by comparing the sigma phase to the dodecagonal phases. The constraint brought the number ratio of triangles T to squares Q to take the value 2.340, very close to that of Stampfli’s quasiperiodic dodecagonal tiling [14] (Fig. 3), √ namely T/Q=4/ 3=2.309. . . . The statistics refer to a sample of 17 000 vertices. Table 1 shows the relative abundancies of vertex configurations extracted from a sample of 175 000 vertices. The distribution strikingly resembles the vertex frequency distribution of the Stampfli tiling. The slight discrepancies can be easily explained. The square-to-triangle constraint imposes a bias in favor of the vertex 22323 and against the vertices 26 and 32 23 . Moreover, one could view the defective vertices 32 231, 324 1, 323 12 and 322 122 (which contain the 30◦ lozenge) as well as the excess of 22323 as defective versions of the missing 26 and 32 23 . It should be noted that real structures practically always contain such defective tiles and vertices. Thus, this vice of the simulation eventually might turn out to be its virtue. With some hindsight, one should not be surprised by the effectiveness of the square-to-triangle rule or by its bias, since it is strongly enforced in the closely related sigma phases, which contain only the vertex 22 323.Within the framework of this research, we have also calculated the exact relative vertex frequencies of the quasiperiodic square–triangle tiling (with randomly oriented dodecagons). These are quoted for comparison in Table 1. The results are as follows:
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B. Rubinstein, S.I. Ben-Abraham / Materials Science and Engineering 294–296 (2000) 418–420
Table 1 Abundancies of vertex configurations in simulated tiling compared to exact vertex frequencies of Stampfli’s dodecagonal square–triangle tilinga
Vertex type Simulated abundance Calculated frequency a
33321 0 0
33231 0.006 0
322221 0.036 0
322212 0.066 0
322122 0.07 0
3333 0 0
33222 0.019 0.091
22323 0.847 0.837
222222 0.016 0.072
The data were extracted from a sample of 17 000 vertices.
√ √ n(22323)=(2 − 3)(7 3 − 9)=0.837 168 574 . . . ≈ 0.84, √ √ n(32 23 )=(2 − 3)(9 − 5 3) = 0.091 034 655 . . . ≈ 0.09, √ √ n(26 ) = (2 − 3)(2 − 3) = 0.071 796 769 . . . ≈ 0.07, n(V) denotes the relative frequency of vertex V. The details of this calculation will be published elsewhere.
3. Conclusions and outlook The simulation showed that the square-to-triangle rule produces realistic structures. Hence we conclude that it reflects genuine chemical interactions between relatively stable atomic clusters. This claim is strongly supported by experimental results on Ta–Te [12]. We are now studying these clusters and their interactions. We are also in the process of significantly scaling up the size of the simulation.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
P. Gummelt, Geom. Dedicata 62 (1996) 1. H.-C. Jeong, P.J. Steinhardt, Phys. Rev. Lett. 73 (1994) 1943. M. Dzugutov, Phys. Rev. A 46 (1992) R2984. V.E. Dmitrienko, S.B. Astaf’ev, Phys. Rev. B 59 (1999) 286. A. Quandt, M.P. Teter, Phys. Rev. B 59 (1999) 8586. D. Joseph, V. Elser, Phys. Rev. Lett. 79 (1997) 1066. D. Joseph, Phys. Rev. B 58 (1998) 8347. T. Ishimasa, H.U. Nissen, Y. Fukano, Phys. Rev. Lett. 55 (1985) 511. F. Gähler, in: C. Janot, J.M. Dubois (Eds.), Quasicrystalline Materials, World Scientific, Singapore, 1988, pp. 272–284. N. Niizeki, H. Mitani, J. Phys. A 20 (1987) 405. H. Chen, D.X. Li, K.H. Kuo, Phys. Rev. Lett. 60 (1988) 1645. M. Conrad, Tantalreiche telluride, Ph.D. Thesis, Dortmund University, Shaker Verlag, Aachen, 1997. K.H. Kuo, Y.C. Feng, H. Chen, Phys. Rev. Lett. 61 (1988) 1740. P. Stampfli, Helv. Phys. Acta 59 (1986) 1260–1263. T. Ishimasa, H.U. Nissen, Y. Fukano, Philos. Mag. A 58 (1988) 835.
The ‘random’ square–triangle tiling: simulation of growth Boaz Rubinstein∗ , Shelomo I Ben-Abraham Department of Physics, Ben-Gurion University of the Negev, PO Box 653, IL-84105 Beer-Sheba, Israel Received 3 September 1999; accepted 6 October 1999
Abstract Some alloy systems, such as Ni–Cr, V–Ni–Si and Ta–Te, have quasicrystalline phases with 12-fold symmetry. These structures may be described in terms of dodecagonal tilings by equilateral triangles and squares. The formation of quasicrystals still poses a problem, since local information is insufficient for the construction of a perfect quasiperiodic structure. The growth of real quasicrystals may be due to several mechanisms. We have simulated the growth of a quasicrystal from a melt, consisting of squares and equilateral triangles of equal edge length. We are interested in the abundancies of the vertex configurations formed, both regular and defective. Unrestricted random growth tends to result in segregation of triangles from squares. Favoring triangles to attract squares and vice versa brings about nearly perfect patterns with nearly perfect vertex abundancies, as well as realistic defect concentrations. We have also calculated the exact vertex frequencies of the ideal square–triangle tiling by relying on inflation symmetry. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Dodecagonal tilings; Vertex; Inflation symmetry
1. Introduction Strictly local information is insufficient for the construction of a perfect generic quasiperiodic structure. Hence it is of continuing interest to study the growth mechanisms of real quasicrystals. One plausible scenario is building up the quasicrystal from some relatively stable structural units pre-existing in the melt through some suitable local interactions. The resulting structure can be abstractly represented by a tiling with some degree of randomness. Its formation can be viewed as a successive agglomeration of building blocks under some more or less stringent constraints. One approach is joining tiles from a given set face to face. An alternative is covering space by partially overlapping clusters [1–2]. According to a complementary model the crystal solidifies from the melt by adding single atoms, as opposed to clusters. The nucleus is allowed to freeze and remelt until equilibrium is reached (cf. [3–7]). In reality, different scenarios may apply to different systems. However, it is plausible that within the same system several mechanisms may compete, or more likely, act in synergy. We try to study these topics by simulating various growth mechanisms. Here we present a simulation of constrained random growth of the dodecagonal square–triangle tiling. The latter is an adequate abstract representation of the do-
∗ Corresponding author. Tel.: +972-7-6472419; fax: +972-7-6472903. E-mail address: [email protected] (B. Rubinstein).
decagonal phases observed in such alloy systems as Ni–Cr [8–10,15], V–Ni and V–Ni–Si [11], Ta–Te [12].
2. Procedure and results The rationale of our simulation is to study the growth of a dodecagonal quasicrystal from a melt. In order to represent reality, the growth should be visualized as an aggregation of pre-existing complex building blocks (clusters) rather than of individual atoms. That, of course, neither explicitly shows up in the tiling model nor in the simulation which is based upon it. The simulation proceeds according to the following simple algorithm: chose an edge at random on the free surface of the growing nucleus and attach a tile to it according to some predetermined rules. If the tile does not overlap other tiles, and is allowed by the algorithm, let it stay, else erase it. Then return to the first step. For technical reasons the whole structure is contained within a rectangular box of area 400 square units, the unit being an edge length. To mimic the chemistry we have tried a variety of restrictions. Among those were forbidding crystallographic vertices, and using “house-shaped” supertiles as building blocks. These consisted of a triangle attached onto a square. We have also varied the “chemical composition” of the melt, that is the ratio of triangles to squares in the melt. Unrestricted growth tended to produce phase separation between domains containing only triangles from domains
0921-5093/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 1 1 4 2 - 4
B. Rubinstein, S.I. Ben-Abraham / Materials Science and Engineering 294–296 (2000) 418–420
419
Fig. 3. Stampfli’s dodecagonal square–triangle tiling. Fig. 1. Phase √ separation in solidification from a melt, of square-to-triangle ratio of 3/4.
of squares, both, of course, periodic by themselves (Fig. 1). Most variations of the growth rules were not particularly effective. However, the simple device of forcing triangles to join squares and vice versa brought about a dramatic improvement. Under this constraint, if the chosen edge belongs to a triangle then attach a square, and vice versa. For example, the vertex 33222 (the digits code for the angles around
Fig. 2. Tiling produced by simulation of growth under the squareto-triangle constraint.
the vertex in units of 2π/12=30◦ ) cannot be built alone, and must be embedded in a larger structure; hence it indeed occurs rather sparsely. The resulting tilings showed a strong preference for local patterns occurring in nature (Fig. 2). A very similar observation was made by Kuo et al. [13], who arrived at their constraint by comparing the sigma phase to the dodecagonal phases. The constraint brought the number ratio of triangles T to squares Q to take the value 2.340, very close to that of Stampfli’s quasiperiodic dodecagonal tiling [14] (Fig. 3), √ namely T/Q=4/ 3=2.309. . . . The statistics refer to a sample of 17 000 vertices. Table 1 shows the relative abundancies of vertex configurations extracted from a sample of 175 000 vertices. The distribution strikingly resembles the vertex frequency distribution of the Stampfli tiling. The slight discrepancies can be easily explained. The square-to-triangle constraint imposes a bias in favor of the vertex 22323 and against the vertices 26 and 32 23 . Moreover, one could view the defective vertices 32 231, 324 1, 323 12 and 322 122 (which contain the 30◦ lozenge) as well as the excess of 22323 as defective versions of the missing 26 and 32 23 . It should be noted that real structures practically always contain such defective tiles and vertices. Thus, this vice of the simulation eventually might turn out to be its virtue. With some hindsight, one should not be surprised by the effectiveness of the square-to-triangle rule or by its bias, since it is strongly enforced in the closely related sigma phases, which contain only the vertex 22 323.Within the framework of this research, we have also calculated the exact relative vertex frequencies of the quasiperiodic square–triangle tiling (with randomly oriented dodecagons). These are quoted for comparison in Table 1. The results are as follows:
420
B. Rubinstein, S.I. Ben-Abraham / Materials Science and Engineering 294–296 (2000) 418–420
Table 1 Abundancies of vertex configurations in simulated tiling compared to exact vertex frequencies of Stampfli’s dodecagonal square–triangle tilinga
Vertex type Simulated abundance Calculated frequency a
33321 0 0
33231 0.006 0
322221 0.036 0
322212 0.066 0
322122 0.07 0
3333 0 0
33222 0.019 0.091
22323 0.847 0.837
222222 0.016 0.072
The data were extracted from a sample of 17 000 vertices.
√ √ n(22323)=(2 − 3)(7 3 − 9)=0.837 168 574 . . . ≈ 0.84, √ √ n(32 23 )=(2 − 3)(9 − 5 3) = 0.091 034 655 . . . ≈ 0.09, √ √ n(26 ) = (2 − 3)(2 − 3) = 0.071 796 769 . . . ≈ 0.07, n(V) denotes the relative frequency of vertex V. The details of this calculation will be published elsewhere.
3. Conclusions and outlook The simulation showed that the square-to-triangle rule produces realistic structures. Hence we conclude that it reflects genuine chemical interactions between relatively stable atomic clusters. This claim is strongly supported by experimental results on Ta–Te [12]. We are now studying these clusters and their interactions. We are also in the process of significantly scaling up the size of the simulation.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
P. Gummelt, Geom. Dedicata 62 (1996) 1. H.-C. Jeong, P.J. Steinhardt, Phys. Rev. Lett. 73 (1994) 1943. M. Dzugutov, Phys. Rev. A 46 (1992) R2984. V.E. Dmitrienko, S.B. Astaf’ev, Phys. Rev. B 59 (1999) 286. A. Quandt, M.P. Teter, Phys. Rev. B 59 (1999) 8586. D. Joseph, V. Elser, Phys. Rev. Lett. 79 (1997) 1066. D. Joseph, Phys. Rev. B 58 (1998) 8347. T. Ishimasa, H.U. Nissen, Y. Fukano, Phys. Rev. Lett. 55 (1985) 511. F. Gähler, in: C. Janot, J.M. Dubois (Eds.), Quasicrystalline Materials, World Scientific, Singapore, 1988, pp. 272–284. N. Niizeki, H. Mitani, J. Phys. A 20 (1987) 405. H. Chen, D.X. Li, K.H. Kuo, Phys. Rev. Lett. 60 (1988) 1645. M. Conrad, Tantalreiche telluride, Ph.D. Thesis, Dortmund University, Shaker Verlag, Aachen, 1997. K.H. Kuo, Y.C. Feng, H. Chen, Phys. Rev. Lett. 61 (1988) 1740. P. Stampfli, Helv. Phys. Acta 59 (1986) 1260–1263. T. Ishimasa, H.U. Nissen, Y. Fukano, Philos. Mag. A 58 (1988) 835.