- Email: [email protected]
Random tiling approach to the structure of decagonal quasicrystals
Journal of Non-Crystalline Solids 153&154 (1993) 253-257 North-Holland
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Random tiling approach to the structure of decagonal quasicrystals Mincheol Shin a and Katherine J. Strandburg b a Department o f Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA b Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA
The properties of random tiling models for the structure of decagonal quasicrystals are studied using computer simulation. In these models the individual layers are random tilings or binary random tilings and the tiling configurations of adjacent layers are coupled. We simulate the equilibrium behavior of the model and compute elastic constants and entropy. We use of the atomic structure models to decorate the tilings and produce simulated electron micrographs and compare them with the images obtained in high-resolution electron microscopy (HREM) experiments. We also compute the diffraction intensities.
1. Introduction
The random tiling or entropic model for quasicrystal structure is based on the suggestion that it is the entropy available from the many, locally similar, rearrangements of the fundamental structural units (or 'tiles') comparising the quasiperiodic structure which stabilizes quasicrystals. One difficulty in gauging the success of this model has been the paucity of direct experimental predictions based on the model. The decagonal quasicrystal AICuCo provides a good testing ground for these models because the current structural models for AICuCo are based quite directly on the two-dimensional (2D) Penrose tiling. Decagonal quasicrystals are quasicrystalline in 2D layers and show ordinary crystalline diffraction perpendicular to the layers. The designs of structural models are generally based on an idealized picture of identical quasiperiodic layers. However, equally possible would be a model of the decagonal phase based on a stack of distinct random tiling layers. Indeed, it is clearly not appropriate to stack and repeat identical 2D random tilings in the stacking direction. The entropy of such a Correspondence to: Dr K.J. Strandburg, Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA.
model is proportional to the layer area of the 2D tilings, whereas the entropy of a 3D system must be proportional to the volume. Thus there must be randomness in the stacking direction if the decagonal phase is to be entropically stabilized. In this work we simulate a stack of interacting tilings introducing the randomness by allowing flips of disjoint hexagon or octagons of adjacent layers [1], as will be described in detail in section 2. We mainly measure the following quantities: (1) the elastic constants; (2) diffraction patterns; (3) simulated electron microscopy images; and (4) entropy.
2. The model
Our model system is three-dimensional stacked axial random tilings. Each layer is a 2D random tiling and two adjacent layers are different only by possible disjointedly flipped hexagons or octagons. A disjoint hexagon or octagon flip is defined in the following way. Given a tile configuration in one layer, one can choose a set of hexagons that do not share any tiles with each other. The adjacent layers, one layer above and below, are identical to the layer in the middle but the hexagons in the chosen set may be flipped (fig. 1). The weight among all the possible sets of disjoint
0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
M. Shin, K.J. Strandburg / The structure of decagonal quasicrystals
254
an average linear phason strain but the strain gets smaller as we take better approximants, i.e. larger systems. The effect of the phason strains are taken into account when we analyze our results. We repeat and stack up the 2D tiling in the z-direction and do the disjoint tile flips as described above. We study the time evolution of the width of the phason fluctuations and also the time-time correlation of the Fourier transform of the perpendicular space coordinates to determine whether the system has been equilibrated. After equilibrium is reached, we take data. Fig. 1. An example of two layers coupled by disjoint hexagon flips. Solid and dashed lines represent tiles in the two layers, respectively.
3. Results
hexagons is equal. An energy cost J is given to each flip between layers and the probability of such a flip is exp(-J/kBT), where T is temperature. We assign no energy costs to matching rule violations induced by the hexagon or octagon flips within a layer because we want to focus on the effect of coupling between the layers. We have simulated two kinds of rhombus tilings: unrestricted and binary tilings. The procedure of our simulation is as follows. We start with a 2D random tiling which is either unrestricted or binary. The 2D tiling is a periodic approximant to a truly quasiperiodic tiling, so that periodic boundary conditions can be implemented. Since it is a periodic approximant, it has
We measured the elastic constants by the Fourier transformation technique [1]. Observing that at all temperatures the elastic constant in the stacking direction, Kz, is much larger than the elastic constant in the plane, Kxy, we investigated the dependence of the elastic constants on the number of layers, L z (fig. 2). For the two temperatures shown in the figure, Kxy is inversely proportional to L~, while K z diverges as Lz increases. This means that the system is so strongly coupled in the stacking direction, even at the infinite temperature, that it effectively behaves as a 2D system and thus prompts us to define a new t ¢ elastic constant Kxy where Kxy =-KxyLz, whose dimensionality is 2D. From the slopes of the curves in fig. 2(a), we get K'y ~ 1.5. We can see
0.20
(a)
12
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'
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8
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1/L
I 0.08
~
I O. 10
0.12
I 20
h 40
I 60
I 80
I 100
120
L
Fig. 2. (a) Plot of Kxy versus 1 / L z. (b) Plot of K z versus L z. The solid lines represent the system at T = 1.0 and the dotted lines at T =c~. Both systems have 199 vertices in each layer. The circles are the elastic constants measured at p = 27r/Lxy, and the triangles at p = 4~r/L~y, where Lxy is the linear dimension of each layer.
M. Shin, K.J. Strandburg / The structure of decagonal quasicrystals
255
Table 1 T h e value of K'y measured at five different peaks. G denotes reciprocal lattice vectors in the 6D notation and G ± their perp-space components. 15 and 16 are the intensities of the systems of 100 layers, with 199 vertices and 521 vertices in each layer, rt ¢ and Kxy are defined in the text G (3, (4, (1, (2, (3,
0, 0, 1, 1, 2,
-4, - 7, - 1, - 1, -2,
0, 0) - 7, 0, - 1, 0, - 2, 0, - 3 , 0,
0) 0) 0) 0)
JG ± I
Log(15)
Log(16)
1 - "q/2
Kjy
1.326 0.820 1.826 1.128 0.697
3.784 4.105 3.313 3.936 4.162
4.164 4.509 3.656 4.322 4.569
0.911 0.966 0.821 0.922 0.974
1.57 1.57 1.48 1.30 1.52)
in the diffraction pattern study that K'y plays the same role in this strongly coupled layered system as Ken, the elastic constant of 2D random tilings, does in 2D random tilings. Namely, we observe that for our layered systems we have power-law diffraction peaks with I(Q) ~ N 1 - n / 2 , where 77 = I G x I 2 / 2 w K ' y . G . is the perp-space momentum vector and N is the total number of vertices in the system. By fitting the peak intensities to the form (table 1), we get K'y ~ 1.5, which is the same as the value obtained from the elastic constant measurement by Fourier transformation. Note that for 2D random tilings, we have the same power-law peak intensities with the elastic constant K2D ( ~ 0.6)replacing K'y in the peak intensity scaling form. Since the system is effec-
2.0
(a)
I
"
'
I
I
tively 2D even for the maximally random tiling of our model, i.e. at the infinite temperature, we can deduce that at all the temperatures in our model, the systems are effectively 2D. The diffuse scatterings in the plane around the peaks also closely resemble 2D random tiling diffuse scatterings (fig. 3(a)), i.e. I(QI I + qxy)~ q-2+n The diffuse scatterings in the z-direction xy can be fit to the form I(QII + qz) ~ q z 2 ( f i g . 3(b)). We produce the simulated H R E M images by decorating our binary tilings [2] and project the 'electron densities' assigned to each atomic position onto a plane perpendicular to the stacking direction. On those images (fig. 4(a)), we can find that clear decagonal rings, which look very similar to the ones seen in H R E M experiments [3], start
I
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I -10
r
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k 5
I 0
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1.0
~
-0.5
0.5
~.~
-1.5
-2.5
0.0 •
/i -3.5
-0.5 20
21
22
23
Log(Q)
24
25
20
20
Log(Q)
Fig. 3. (a) Diffuse scattering at T = o~ (dotted line) and a T = 1.0 (dashed line) with m o m e n t u m vectors in the planar direction for the systems of 50 layers with 199 vertices in each layer. Two nearby peaks are shown in the figure and N is the total n u m b e r of vertices in the system. The diffuse scattering of 2D r a n d o m tilings of the same area is shown for comparison (solid line). (b) Diffuse scattering at T = oo (dotted line) and at T = 1.0 (solid line) with QII - GII in the stacking direction for the systems of the same size. QII is the m o m e n t u m vector and GII is a reciprocal lattice vector. In the figure, G = (1, 0, - 1, - 1, 0, 0) in the 6D notation and the projection of G onto the physical space gives GII.
M. Shin, K.J. Strandburg / The structure of decagonal quasicrystals
256
~
• .
-
~
Fig. 4. (a) A simulated H R E M image at T = 2.0 with a part of the region tiled by two rings, (b) The reconstructed tiling of the entire region of part (a).
to show up for systems with more than about 15 stacked layers. We connect the rings to reconstruct new tilings (fig. 4(b)) and plot the perpspace profiles of the new filings (fig. 5). Note the similarity between the reconstructed tiling of a H R E M image from experiments (fig. 6) and that
• .
.:~.
from our simulation. Also not that our individual layers are random tilings whose projected images do not yield the decagonal rings. It is interesting
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Fig. 5. Perp-space profile of a reconstructed tiling of the fig. ~b).
Fig. 6. Reconstructed tiling of a H R E M image from experiments (courtesy of Goldman et al.).
M. Shin, K.J. Strandburg / The structure of decagonal quasicrystals
257
4. Conclusion
0.25
j
0.20
0.15
f
....
.o
b~ 0.10
0.05
,
0.(~0 10
,
,
,
,
,
,
,]
,
,
,
,
10 °
101
Log(T) Fig. 7. Plot of the entropy versus temperature. to see that upon stacking up many such random tilings, the decagonal rings show up as the average image of the layers and the rings are positioned in such a way that one can tile the entire region in a consistent way and its perp-space looks closer to that of the ideal tiling than that of random tilings. Similar behavior has been observed by Shaw et al. [4] in a simulation of icosahedral quasicrystals; they observed a crossover to the region of almost absent phason fluctuation when the thickness of their model system was increased. We also measure the entropy by first measuring the energy E ( T ) as a function of T (fig. 7). The entropy at the infinite t e m p e r a t u r e is 0.210. Note that this value is lower than that of 2D random tilings, 0.32, which is expected because the coupling between layers of our model systems constrains the possible n u m b e r of tile configurations.
O u r model implemented by the disjoint hexagon flips (see section 2) yields strongly coupled layered systems which behave effectively as 2D random tilings in many respects. We think that the geometric constraint that two adjacent layers are the same except at the centers of disjoint hexagons is rather too strong, even for the maximally random tiling of the model. This model, however, can be regarded as the guide for the future work. In fact, it led us to devise another model where the strength of the coupling between the layers can be varied from 0 to infinity. However, for wide range of the strength of the coupling of the new model, we have found that the systems are also effectively 2D. We will report the detailed results elsewhere. We thank C.L. Henley for useful discussions and A.I. Goldman and L.X. He for providing us with the H R E M micrographs. This work was supported by the U.S. D e p a r t m e n t of Energy, BESMaterials Sciences, under Contract No. W-31109-ENG-38.
References [1] C.L. Henley, Quasicrystals: The State of the Art ed. D.P. DiVincenzo and P.J. Steinhardt (World Scientific, Singapore, 1991) p. 429. [2] We used the Neo-Burkov model, see C.L. Henley, J. Non-Cryst. Solids 153&154 (1993) 172. [3] A.I. Goldman, private communication. [4] L.J. Shaw, V. Elser and C.L. Henley, Phys. Rev. B43 (1991) 3423.
joueNA ~ or
~
~
U
~
Random tiling approach to the structure of decagonal quasicrystals Mincheol Shin a and Katherine J. Strandburg b a Department o f Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA b Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA
The properties of random tiling models for the structure of decagonal quasicrystals are studied using computer simulation. In these models the individual layers are random tilings or binary random tilings and the tiling configurations of adjacent layers are coupled. We simulate the equilibrium behavior of the model and compute elastic constants and entropy. We use of the atomic structure models to decorate the tilings and produce simulated electron micrographs and compare them with the images obtained in high-resolution electron microscopy (HREM) experiments. We also compute the diffraction intensities.
1. Introduction
The random tiling or entropic model for quasicrystal structure is based on the suggestion that it is the entropy available from the many, locally similar, rearrangements of the fundamental structural units (or 'tiles') comparising the quasiperiodic structure which stabilizes quasicrystals. One difficulty in gauging the success of this model has been the paucity of direct experimental predictions based on the model. The decagonal quasicrystal AICuCo provides a good testing ground for these models because the current structural models for AICuCo are based quite directly on the two-dimensional (2D) Penrose tiling. Decagonal quasicrystals are quasicrystalline in 2D layers and show ordinary crystalline diffraction perpendicular to the layers. The designs of structural models are generally based on an idealized picture of identical quasiperiodic layers. However, equally possible would be a model of the decagonal phase based on a stack of distinct random tiling layers. Indeed, it is clearly not appropriate to stack and repeat identical 2D random tilings in the stacking direction. The entropy of such a Correspondence to: Dr K.J. Strandburg, Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA.
model is proportional to the layer area of the 2D tilings, whereas the entropy of a 3D system must be proportional to the volume. Thus there must be randomness in the stacking direction if the decagonal phase is to be entropically stabilized. In this work we simulate a stack of interacting tilings introducing the randomness by allowing flips of disjoint hexagon or octagons of adjacent layers [1], as will be described in detail in section 2. We mainly measure the following quantities: (1) the elastic constants; (2) diffraction patterns; (3) simulated electron microscopy images; and (4) entropy.
2. The model
Our model system is three-dimensional stacked axial random tilings. Each layer is a 2D random tiling and two adjacent layers are different only by possible disjointedly flipped hexagons or octagons. A disjoint hexagon or octagon flip is defined in the following way. Given a tile configuration in one layer, one can choose a set of hexagons that do not share any tiles with each other. The adjacent layers, one layer above and below, are identical to the layer in the middle but the hexagons in the chosen set may be flipped (fig. 1). The weight among all the possible sets of disjoint
0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
M. Shin, K.J. Strandburg / The structure of decagonal quasicrystals
254
an average linear phason strain but the strain gets smaller as we take better approximants, i.e. larger systems. The effect of the phason strains are taken into account when we analyze our results. We repeat and stack up the 2D tiling in the z-direction and do the disjoint tile flips as described above. We study the time evolution of the width of the phason fluctuations and also the time-time correlation of the Fourier transform of the perpendicular space coordinates to determine whether the system has been equilibrated. After equilibrium is reached, we take data. Fig. 1. An example of two layers coupled by disjoint hexagon flips. Solid and dashed lines represent tiles in the two layers, respectively.
3. Results
hexagons is equal. An energy cost J is given to each flip between layers and the probability of such a flip is exp(-J/kBT), where T is temperature. We assign no energy costs to matching rule violations induced by the hexagon or octagon flips within a layer because we want to focus on the effect of coupling between the layers. We have simulated two kinds of rhombus tilings: unrestricted and binary tilings. The procedure of our simulation is as follows. We start with a 2D random tiling which is either unrestricted or binary. The 2D tiling is a periodic approximant to a truly quasiperiodic tiling, so that periodic boundary conditions can be implemented. Since it is a periodic approximant, it has
We measured the elastic constants by the Fourier transformation technique [1]. Observing that at all temperatures the elastic constant in the stacking direction, Kz, is much larger than the elastic constant in the plane, Kxy, we investigated the dependence of the elastic constants on the number of layers, L z (fig. 2). For the two temperatures shown in the figure, Kxy is inversely proportional to L~, while K z diverges as Lz increases. This means that the system is so strongly coupled in the stacking direction, even at the infinite temperature, that it effectively behaves as a 2D system and thus prompts us to define a new t ¢ elastic constant Kxy where Kxy =-KxyLz, whose dimensionality is 2D. From the slopes of the curves in fig. 2(a), we get K'y ~ 1.5. We can see
0.20
(a)
12
I
(bi
'
'
i
I
,
10 0.15
..-"
.D ......-":"
8
~I~~
0.10
6 4
0.05
0.00 0.00
I 0.02
b 0.04
I 0.06
1/L
I 0.08
~
I O. 10
0.12
I 20
h 40
I 60
I 80
I 100
120
L
Fig. 2. (a) Plot of Kxy versus 1 / L z. (b) Plot of K z versus L z. The solid lines represent the system at T = 1.0 and the dotted lines at T =c~. Both systems have 199 vertices in each layer. The circles are the elastic constants measured at p = 27r/Lxy, and the triangles at p = 4~r/L~y, where Lxy is the linear dimension of each layer.
M. Shin, K.J. Strandburg / The structure of decagonal quasicrystals
255
Table 1 T h e value of K'y measured at five different peaks. G denotes reciprocal lattice vectors in the 6D notation and G ± their perp-space components. 15 and 16 are the intensities of the systems of 100 layers, with 199 vertices and 521 vertices in each layer, rt ¢ and Kxy are defined in the text G (3, (4, (1, (2, (3,
0, 0, 1, 1, 2,
-4, - 7, - 1, - 1, -2,
0, 0) - 7, 0, - 1, 0, - 2, 0, - 3 , 0,
0) 0) 0) 0)
JG ± I
Log(15)
Log(16)
1 - "q/2
Kjy
1.326 0.820 1.826 1.128 0.697
3.784 4.105 3.313 3.936 4.162
4.164 4.509 3.656 4.322 4.569
0.911 0.966 0.821 0.922 0.974
1.57 1.57 1.48 1.30 1.52)
in the diffraction pattern study that K'y plays the same role in this strongly coupled layered system as Ken, the elastic constant of 2D random tilings, does in 2D random tilings. Namely, we observe that for our layered systems we have power-law diffraction peaks with I(Q) ~ N 1 - n / 2 , where 77 = I G x I 2 / 2 w K ' y . G . is the perp-space momentum vector and N is the total number of vertices in the system. By fitting the peak intensities to the form (table 1), we get K'y ~ 1.5, which is the same as the value obtained from the elastic constant measurement by Fourier transformation. Note that for 2D random tilings, we have the same power-law peak intensities with the elastic constant K2D ( ~ 0.6)replacing K'y in the peak intensity scaling form. Since the system is effec-
2.0
(a)
I
"
'
I
I
tively 2D even for the maximally random tiling of our model, i.e. at the infinite temperature, we can deduce that at all the temperatures in our model, the systems are effectively 2D. The diffuse scatterings in the plane around the peaks also closely resemble 2D random tiling diffuse scatterings (fig. 3(a)), i.e. I(QI I + qxy)~ q-2+n The diffuse scatterings in the z-direction xy can be fit to the form I(QII + qz) ~ q z 2 ( f i g . 3(b)). We produce the simulated H R E M images by decorating our binary tilings [2] and project the 'electron densities' assigned to each atomic position onto a plane perpendicular to the stacking direction. On those images (fig. 4(a)), we can find that clear decagonal rings, which look very similar to the ones seen in H R E M experiments [3], start
I
I"
I -15
I -10
r
I
I
I
]
I
k 5
I 0
I 5
I 10
I 15
0.5
1.5
O
I
(b)
1.0
~
-0.5
0.5
~.~
-1.5
-2.5
0.0 •
/i -3.5
-0.5 20
21
22
23
Log(Q)
24
25
20
20
Log(Q)
Fig. 3. (a) Diffuse scattering at T = o~ (dotted line) and a T = 1.0 (dashed line) with m o m e n t u m vectors in the planar direction for the systems of 50 layers with 199 vertices in each layer. Two nearby peaks are shown in the figure and N is the total n u m b e r of vertices in the system. The diffuse scattering of 2D r a n d o m tilings of the same area is shown for comparison (solid line). (b) Diffuse scattering at T = oo (dotted line) and at T = 1.0 (solid line) with QII - GII in the stacking direction for the systems of the same size. QII is the m o m e n t u m vector and GII is a reciprocal lattice vector. In the figure, G = (1, 0, - 1, - 1, 0, 0) in the 6D notation and the projection of G onto the physical space gives GII.
M. Shin, K.J. Strandburg / The structure of decagonal quasicrystals
256
~
• .
-
~
Fig. 4. (a) A simulated H R E M image at T = 2.0 with a part of the region tiled by two rings, (b) The reconstructed tiling of the entire region of part (a).
to show up for systems with more than about 15 stacked layers. We connect the rings to reconstruct new tilings (fig. 4(b)) and plot the perpspace profiles of the new filings (fig. 5). Note the similarity between the reconstructed tiling of a H R E M image from experiments (fig. 6) and that
• .
.:~.
from our simulation. Also not that our individual layers are random tilings whose projected images do not yield the decagonal rings. It is interesting
.. o e
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.
•
.
•
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Fig. 5. Perp-space profile of a reconstructed tiling of the fig. ~b).
Fig. 6. Reconstructed tiling of a H R E M image from experiments (courtesy of Goldman et al.).
M. Shin, K.J. Strandburg / The structure of decagonal quasicrystals
257
4. Conclusion
0.25
j
0.20
0.15
f
....
.o
b~ 0.10
0.05
,
0.(~0 10
,
,
,
,
,
,
,]
,
,
,
,
10 °
101
Log(T) Fig. 7. Plot of the entropy versus temperature. to see that upon stacking up many such random tilings, the decagonal rings show up as the average image of the layers and the rings are positioned in such a way that one can tile the entire region in a consistent way and its perp-space looks closer to that of the ideal tiling than that of random tilings. Similar behavior has been observed by Shaw et al. [4] in a simulation of icosahedral quasicrystals; they observed a crossover to the region of almost absent phason fluctuation when the thickness of their model system was increased. We also measure the entropy by first measuring the energy E ( T ) as a function of T (fig. 7). The entropy at the infinite t e m p e r a t u r e is 0.210. Note that this value is lower than that of 2D random tilings, 0.32, which is expected because the coupling between layers of our model systems constrains the possible n u m b e r of tile configurations.
O u r model implemented by the disjoint hexagon flips (see section 2) yields strongly coupled layered systems which behave effectively as 2D random tilings in many respects. We think that the geometric constraint that two adjacent layers are the same except at the centers of disjoint hexagons is rather too strong, even for the maximally random tiling of the model. This model, however, can be regarded as the guide for the future work. In fact, it led us to devise another model where the strength of the coupling between the layers can be varied from 0 to infinity. However, for wide range of the strength of the coupling of the new model, we have found that the systems are also effectively 2D. We will report the detailed results elsewhere. We thank C.L. Henley for useful discussions and A.I. Goldman and L.X. He for providing us with the H R E M micrographs. This work was supported by the U.S. D e p a r t m e n t of Energy, BESMaterials Sciences, under Contract No. W-31109-ENG-38.
References [1] C.L. Henley, Quasicrystals: The State of the Art ed. D.P. DiVincenzo and P.J. Steinhardt (World Scientific, Singapore, 1991) p. 429. [2] We used the Neo-Burkov model, see C.L. Henley, J. Non-Cryst. Solids 153&154 (1993) 172. [3] A.I. Goldman, private communication. [4] L.J. Shaw, V. Elser and C.L. Henley, Phys. Rev. B43 (1991) 3423.