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Percolation thresholds and percolation conductivities of octagonal and dodecagonal quasicrystalline lattices
PHYSICA/ ELSEVIER
Physica A 220 (1995) 245-250
Percolation thresholds and percolation conductivities of octagonal and dodecagonal quasicrystalline lattices F. Babalievski Institute of General and Inorganic Chemistry, 1113 Sofia, Bulgaria
Received 23 May 1995; revised 5 August 1995
Abstract
The octagonal and dodecagonal quasilattices were generated by means of the grid method. Monte Carlo simulation and cluster counting procedure were used for numerical determination of the site and bond percolation thresholds. Two types of connectivity called ferromagnetic and chemical were studied. The estimated site percolation thresholds are 0.5435... and 0.585... for octagonal lattice and 0.617... and 0.628... for dodecagonal lattice respectively. The obtained spanning fraction curves(for site percolation) seem to approach the 50% value at the percolation threshold. The site percolation conductivity for these lattices was studied by means of a transfermatrix approach. The critical behavior was found to be the same as for the periodic lattices.
Several decades after introducing the percolation theory, a simple connection between the percolation threshold and the other properties of a lattice still cannot be found. There are few exceptions where the percolation threshold is exactly known. A lot of attempts were made to establish at least approximate invariants which include the percolation threshold [ 1,3,4]. As described below, it is difficult to apply these invariants for the case of aperiodic lattices with long range order - the quasicrystalline lattices [2]. A better knowledge of the percolation properties of quasicrystalline lattices with different symmetries is important itself and could help finding such invartiants. Indeed, the percolation characteristics are important for every lattice structure which can serve as a model for an alloy-like material: at least one has to know what is the concentration for which an infinite cluster of nearest neighbors of identical atoms spans the sample the percolation threshold. 0378-4371/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0378-4371 (95)00260-X
246
F. Babalievski/Physica A 220 (1995) 245-250
That is the reason why this study is mainly devoted to estimating the site percolation thresholds of two of the most important quasilattices: the simple octagonal and dodecagonal lattices [5]. The first studies of percolation models on 2D quasilattices were made [4,6,7] on the Penrose tiling. The aim was one to see if the quasilattices belong to the same universality class as the percolation models on periodic lattices. It became clear that, within the estimations error, the critical exponents are the same. In order to study critical exponents for a percolation model one has to know the percolation threshold, so the percolation threshold for site and bond percolation on Penrose lattice and its dual were estimated with due precision. It was attempted to compare this set of percolation thresholds with the values predicted by the threshold invariants already known for periodic and random lattices. I will discuss here only the site percolation problem ( for bond percolation on Penrose tiling see Ref. [4] ). In Ref. [7], the Scher and Zallen approximate invariant was used for comparison with Monte Carlo simulation results for the site percolation threshold of the Penrose tiling and its dual lattice. As known, the Scher and Zallen approximate relation is: P c f ,~ 0.45
in two dimensions ( f is the filling factor for the respective lattice). The results did not convince us that this invariant could cover the case of quasicrystalline lattices. The reason is that there is not unambiguous definition of the filling factor for a quasilattice. It can not be packed with disks of equal radii which halve the bond length. It was posed that the lattice sites are connected only via the tile edges. Indeed a connection through the diagonals of the tiles which are shorter than the tile edge might exist. Adding such connections leads to models named in Ref. [9] ferromagnetic percolation for evident reasons. On the same ground, the models which use as bonds only the tile edges could be named chemical percolation - the connectivity spreads only through the chemical bonds (the tile's edges) not through sites on shorter distances from each other. The result given in Ref. [9] was 0.54... but it is more probably around 0.57...
[ 10]. So a good definition for the filling factor should distinguish between a ferromagnetic and a chemical variant of connectivity. In the present study , the data for the percolation threshold in the ferromagnetic and chemical case is extended to octagonal and dodecagonal lattices. There is an infinite set of octagonal and dodecagonal tilings which group in local isomoprphism (LI) classes. I used (Figs. lb,d) the simple octagonal and dodecagonal tilings [5] which belong to the Penrose LI class. The lattices were generated by the grid method for creating quasicrystalline structures. For that purpose, the recursive algorithm proposed for the case of Penrose tiling in a previous work [ 11 ] was adopted. Recursive function(i.e, procedure, subroutine, etc. ) was used for finding out all of the meshes(facets) in the "multigrid"(a dual lattice to the respective quasicristalline lattice). The difference in the constructing algorithm for
247
F. Babalievski/Physica A 220 (1995) 245-250
I)
b)
e)
d)
Fig. 1. Pieces of the octagonal quasicrystalline lattice (a) ferromagneticlinks, and (b) chemical links; and for dodecagonal lattices (c) ferromagneticand (d) chemicallinks. the octagonal tiling was only in the definition of the multigrid. An additional problem arose for the dodecagonal tiling. The corresponding "multigrid" is singular: there are points in it, where more than two lines cross each other [5]. The problem was solved by limiting the recursion depth. The computer simulation approach was used for determining the percolation threshold. A modified Hoshen-Kopelman procedure was used for the spanning cluster identification. As described in [ 11] the modification consists in the way the lattice sites' coordinates are handled. The site percolation conductivity of these lattices were studied as well. A modification of the numerical transfer-matrix approach to random resistor network problem [ 12] was used. The algorithm description can be found in [13]. The size of the studied pieces of lattices ranged between 30 and 300 tile edge lengths. For every p (site occupation probability) up to 1000 different configurations were created. For every configuration it was checked if a cluster spans between the left and right edge of the sample. The frequency of such event was estimated as the number of samples spanned, over the number of all samples. Then, for each size, a spanning probability curve was drawn as a function of p. As seen on the Fig. 2 these curves cross the 50% level for almost the same argument. This result supports the conjecture [ 14] that the spanning fraction at the percolation threshold has a value which depends only on the boundary conditions and the spatial dimension. This value is for two dimensions and open bondaries. The percolation thresholds were estimated after inspecting by eye the family of spanning fraction curves for each (lattice)/(type of connectivity) combination (Fig. 2). The estimations for the percolation thresholds are as follows: for chemical links on
E Babalievski/PhysicaA 220 (1995) 245-250
248
octqou.+! (Ch) L=34 ~%.AAAL=I04 L=I70 I..=250
Octqon.l (F) nnnr P L=' 0
1.00
x x x x_._.__x
77
0.50
~:0.54,2 0.~
0.54
0.52
0.56
Dodeeqonal (F) 1.00 ~ L=IO0 I L=200 t31~212CIL=3 I ~ . 7 '
~
/. 0.57
~ = ~ 585 ~=0. $84
0.58
0.59
0.60
Dodeeallonll (Oh) I cil:/a~lL=34 l l ~IL=180I a~ ~ L=2381 ~ . . . . . iL=3ISI ~ l
I" i
0.50
I
t 0"628
g 0.1~
0.60
0.61
0.02:
0.63
0.01
0.02
0.63
0.84
Fig. 2. The spanning fraction curves for: octagonal lattice with (F) ferromagnetic and (Ch) chemical links; and for dodecagonal lattice (F) ferromagnetic and (Ch) chemical links. octagonal lattice pc = 0.585 ± 0.001; for ferromagnetic links and octagonal lattice pc = 0.543 + 0.002; and for dodecagonal lattice the respective values were Pc = 0.628 + 0.002 and Pc = 0.617 i 0.003. The error bars correspond to the uncertainty o f the position where the different curves cross each other. On the other side the mean coordination number Z o f these lattices could be easily obtained ( k n o w i n g the density o f the different tiles which construct the lattice) as the ratio:
~-~i DiVi - ~-~iDiAl'
Ai -
Vi - 2 2 '
where Di is the relative frequency o f the "tile" o f type i; (i = 1 . . . n ) ; n being the number o f different tile types. Vi is the number o f the apices o f the tiles o f type i. (It is assumed that all o f the tile apices form lattice sites as on Fig. 1.) The juxtaposition between ~ and Pc can be seen in Table 1. The third column o f the Table gives the preliminary results for bond percolation. As seen in the last column the approximate relation [15] z/-'c =-bond ,-~ 2 holds within 5%
249
E Babalievski/Physica A 220 (1995) 245-250
Table 1 Juxtaposition between ~ and Pc. Lattice
~
psite
pcbOnd
~pcbOnd
Octagonal lattice, "chemical" links Octagonal lattice, "ferromagnetic"links Dodecagonal lattice, "chemical" links Dodecagonal lattice, "ferromagnetic"links
4 5.17 ... 3.63 ... 4.27 ...
0.585 ... 0.543 ... 0.628... 0.617 ...
0.48 ... 0.40... 0.54 . .. 0.495 . ..
1.92 2.07 1.96 2.11
a)
b) ]~ Q ,
0 . 04
~ e ~ - o Dadeca ,tonal ~-~~ 0 c t a g o l tal
Octagonal
/
/Y
0. 03
~
~
0. 02
0 O.OL .
0.00 0.50
0.00
0.70
0.00
P
0.90
1.00
10
.
.
.
i s i÷.t'
"',,
10 Strip Width
t
s
i
Fig. 3. The transverse conductivityper unit length (a) of strips (30 x 20000 bond(link) lengths) of RRN based on octagonal and dodecagonallattices (chemical links) as function of p; and (b) of strips of different width at the percolationthreshold. deviation. The site percolation conductivity was studied only for the case of "chemical" links by means of the transfer-matrix approach to the random resistor network (RRN) problem. The R R N ' s was built supposing that two "open" lattice sites sharing a bond are connected by a unit resistor. Typical for the transfer-matrix approach is that long and relatively narrow strips from resistors are studied. The transverse conductivity of those strips should be determined for a set of different p and different widths. When p is fixed at p = Pc the scaling of the conductivity with the strip width allows one to estimate the percolation transport exponent for the model studied. The strips of resistors were 20000 bonds long and their width was varied up to 30 bonds. The conductivity scaling found for the both lattices is given on Fig. 3b. No significant deviation from the already known for the other 2D lattices critical behavior was found. The estimation error was around 5-10%: rather large for a reliable value for the percolation transport exponent. The conductivity of strips of fixed width (~-, 30) for p varied in the whole region of nonzero conductivities ( p ~ 0.5-1 ) was obtained as well. The results are given on Fig. 3a.
250
E Babalievski/Physica A 220 (1995) 245-250
In conclusion, the presented spanning curves for quasicrystalline lattices support the (limited) universality for percolation fractions at the percolation threshold in two dimensions. The percolation conductivity of dodecagonal and octagonal latices has the same behavior as the Penrose lattice and the periodic lattices. The estimates for the site percolation thresholds of the octagonal and dodecagonal lattice are given for two ways of linking the sites. These estimates could help in searching analytical expressions for the percolation thresholds. I thank V. Tonchev for useful discussions. This study was supported by the Bulgarian NSF Grant No. F-119/91.
References [ 1] [2] [3] [4] [5] [6] [7] [8] [9] 110] l 11 ] [ 12] [13] [ 14] [15]
H.Scher and R.Zallen, J. Chem. Phys. 53 (1970) 3759. M. Duneau and A. Katz, Phys. Rev. Lett. 54 (1985) 2688. U. Alon, I. Balberg and U. Drory, Phys. Rev. Lett. 66 (1991) 2879. J. P. Lu and J. L Birman, J. Stat. Phys. 40 (1987) 1057. J. Socolar, Phys. Rev. B 39 (1989) 10519. E Babalievski and O. Peshev, C. R. Acad. Sci. Bulg. 41 (1988) 85. E Yonesava, S. Sakamoto, K. Aoki, S. Nose and M. Hod, J. Non-Cryst. Solids 123 (1988) 73. D. Stauffer and A. Aharony, Introduction to percolation theory (Taylor and Francis, London, 1992); second printing 1994. D. Leduie and J. Teillet, unpublished. E Babalievski, unpublished. F. Babalievski and O. Peshev, Comput. Phys. Commun. 60 (1990) 27. B. Derrida, J.G. Zabolitzki, J. Vanimenus and D. Stauffer, J. Stat. Phys. 36 (1984) 31. E Babalievski, Z. Phys. B 84 (1991) 429. U. Gropengiesser and D. Stauffer, Physica A 210 (1994) 320, and references therein. C. Domb and M.E Sykes, Phys. Rev. 122 (1961)77.
Physica A 220 (1995) 245-250
Percolation thresholds and percolation conductivities of octagonal and dodecagonal quasicrystalline lattices F. Babalievski Institute of General and Inorganic Chemistry, 1113 Sofia, Bulgaria
Received 23 May 1995; revised 5 August 1995
Abstract
The octagonal and dodecagonal quasilattices were generated by means of the grid method. Monte Carlo simulation and cluster counting procedure were used for numerical determination of the site and bond percolation thresholds. Two types of connectivity called ferromagnetic and chemical were studied. The estimated site percolation thresholds are 0.5435... and 0.585... for octagonal lattice and 0.617... and 0.628... for dodecagonal lattice respectively. The obtained spanning fraction curves(for site percolation) seem to approach the 50% value at the percolation threshold. The site percolation conductivity for these lattices was studied by means of a transfermatrix approach. The critical behavior was found to be the same as for the periodic lattices.
Several decades after introducing the percolation theory, a simple connection between the percolation threshold and the other properties of a lattice still cannot be found. There are few exceptions where the percolation threshold is exactly known. A lot of attempts were made to establish at least approximate invariants which include the percolation threshold [ 1,3,4]. As described below, it is difficult to apply these invariants for the case of aperiodic lattices with long range order - the quasicrystalline lattices [2]. A better knowledge of the percolation properties of quasicrystalline lattices with different symmetries is important itself and could help finding such invartiants. Indeed, the percolation characteristics are important for every lattice structure which can serve as a model for an alloy-like material: at least one has to know what is the concentration for which an infinite cluster of nearest neighbors of identical atoms spans the sample the percolation threshold. 0378-4371/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0378-4371 (95)00260-X
246
F. Babalievski/Physica A 220 (1995) 245-250
That is the reason why this study is mainly devoted to estimating the site percolation thresholds of two of the most important quasilattices: the simple octagonal and dodecagonal lattices [5]. The first studies of percolation models on 2D quasilattices were made [4,6,7] on the Penrose tiling. The aim was one to see if the quasilattices belong to the same universality class as the percolation models on periodic lattices. It became clear that, within the estimations error, the critical exponents are the same. In order to study critical exponents for a percolation model one has to know the percolation threshold, so the percolation threshold for site and bond percolation on Penrose lattice and its dual were estimated with due precision. It was attempted to compare this set of percolation thresholds with the values predicted by the threshold invariants already known for periodic and random lattices. I will discuss here only the site percolation problem ( for bond percolation on Penrose tiling see Ref. [4] ). In Ref. [7], the Scher and Zallen approximate invariant was used for comparison with Monte Carlo simulation results for the site percolation threshold of the Penrose tiling and its dual lattice. As known, the Scher and Zallen approximate relation is: P c f ,~ 0.45
in two dimensions ( f is the filling factor for the respective lattice). The results did not convince us that this invariant could cover the case of quasicrystalline lattices. The reason is that there is not unambiguous definition of the filling factor for a quasilattice. It can not be packed with disks of equal radii which halve the bond length. It was posed that the lattice sites are connected only via the tile edges. Indeed a connection through the diagonals of the tiles which are shorter than the tile edge might exist. Adding such connections leads to models named in Ref. [9] ferromagnetic percolation for evident reasons. On the same ground, the models which use as bonds only the tile edges could be named chemical percolation - the connectivity spreads only through the chemical bonds (the tile's edges) not through sites on shorter distances from each other. The result given in Ref. [9] was 0.54... but it is more probably around 0.57...
[ 10]. So a good definition for the filling factor should distinguish between a ferromagnetic and a chemical variant of connectivity. In the present study , the data for the percolation threshold in the ferromagnetic and chemical case is extended to octagonal and dodecagonal lattices. There is an infinite set of octagonal and dodecagonal tilings which group in local isomoprphism (LI) classes. I used (Figs. lb,d) the simple octagonal and dodecagonal tilings [5] which belong to the Penrose LI class. The lattices were generated by the grid method for creating quasicrystalline structures. For that purpose, the recursive algorithm proposed for the case of Penrose tiling in a previous work [ 11 ] was adopted. Recursive function(i.e, procedure, subroutine, etc. ) was used for finding out all of the meshes(facets) in the "multigrid"(a dual lattice to the respective quasicristalline lattice). The difference in the constructing algorithm for
247
F. Babalievski/Physica A 220 (1995) 245-250
I)
b)
e)
d)
Fig. 1. Pieces of the octagonal quasicrystalline lattice (a) ferromagneticlinks, and (b) chemical links; and for dodecagonal lattices (c) ferromagneticand (d) chemicallinks. the octagonal tiling was only in the definition of the multigrid. An additional problem arose for the dodecagonal tiling. The corresponding "multigrid" is singular: there are points in it, where more than two lines cross each other [5]. The problem was solved by limiting the recursion depth. The computer simulation approach was used for determining the percolation threshold. A modified Hoshen-Kopelman procedure was used for the spanning cluster identification. As described in [ 11] the modification consists in the way the lattice sites' coordinates are handled. The site percolation conductivity of these lattices were studied as well. A modification of the numerical transfer-matrix approach to random resistor network problem [ 12] was used. The algorithm description can be found in [13]. The size of the studied pieces of lattices ranged between 30 and 300 tile edge lengths. For every p (site occupation probability) up to 1000 different configurations were created. For every configuration it was checked if a cluster spans between the left and right edge of the sample. The frequency of such event was estimated as the number of samples spanned, over the number of all samples. Then, for each size, a spanning probability curve was drawn as a function of p. As seen on the Fig. 2 these curves cross the 50% level for almost the same argument. This result supports the conjecture [ 14] that the spanning fraction at the percolation threshold has a value which depends only on the boundary conditions and the spatial dimension. This value is for two dimensions and open bondaries. The percolation thresholds were estimated after inspecting by eye the family of spanning fraction curves for each (lattice)/(type of connectivity) combination (Fig. 2). The estimations for the percolation thresholds are as follows: for chemical links on
E Babalievski/PhysicaA 220 (1995) 245-250
248
octqou.+! (Ch) L=34 ~%.AAAL=I04 L=I70 I..=250
Octqon.l (F) nnnr P L=' 0
1.00
x x x x_._.__x
77
0.50
~:0.54,2 0.~
0.54
0.52
0.56
Dodeeqonal (F) 1.00 ~ L=IO0 I L=200 t31~212CIL=3 I ~ . 7 '
~
/. 0.57
~ = ~ 585 ~=0. $84
0.58
0.59
0.60
Dodeeallonll (Oh) I cil:/a~lL=34 l l ~IL=180I a~ ~ L=2381 ~ . . . . . iL=3ISI ~ l
I" i
0.50
I
t 0"628
g 0.1~
0.60
0.61
0.02:
0.63
0.01
0.02
0.63
0.84
Fig. 2. The spanning fraction curves for: octagonal lattice with (F) ferromagnetic and (Ch) chemical links; and for dodecagonal lattice (F) ferromagnetic and (Ch) chemical links. octagonal lattice pc = 0.585 ± 0.001; for ferromagnetic links and octagonal lattice pc = 0.543 + 0.002; and for dodecagonal lattice the respective values were Pc = 0.628 + 0.002 and Pc = 0.617 i 0.003. The error bars correspond to the uncertainty o f the position where the different curves cross each other. On the other side the mean coordination number Z o f these lattices could be easily obtained ( k n o w i n g the density o f the different tiles which construct the lattice) as the ratio:
~-~i DiVi - ~-~iDiAl'
Ai -
Vi - 2 2 '
where Di is the relative frequency o f the "tile" o f type i; (i = 1 . . . n ) ; n being the number o f different tile types. Vi is the number o f the apices o f the tiles o f type i. (It is assumed that all o f the tile apices form lattice sites as on Fig. 1.) The juxtaposition between ~ and Pc can be seen in Table 1. The third column o f the Table gives the preliminary results for bond percolation. As seen in the last column the approximate relation [15] z/-'c =-bond ,-~ 2 holds within 5%
249
E Babalievski/Physica A 220 (1995) 245-250
Table 1 Juxtaposition between ~ and Pc. Lattice
~
psite
pcbOnd
~pcbOnd
Octagonal lattice, "chemical" links Octagonal lattice, "ferromagnetic"links Dodecagonal lattice, "chemical" links Dodecagonal lattice, "ferromagnetic"links
4 5.17 ... 3.63 ... 4.27 ...
0.585 ... 0.543 ... 0.628... 0.617 ...
0.48 ... 0.40... 0.54 . .. 0.495 . ..
1.92 2.07 1.96 2.11
a)
b) ]~ Q ,
0 . 04
~ e ~ - o Dadeca ,tonal ~-~~ 0 c t a g o l tal
Octagonal
/
/Y
0. 03
~
~
0. 02
0 O.OL .
0.00 0.50
0.00
0.70
0.00
P
0.90
1.00
10
.
.
.
i s i÷.t'
"',,
10 Strip Width
t
s
i
Fig. 3. The transverse conductivityper unit length (a) of strips (30 x 20000 bond(link) lengths) of RRN based on octagonal and dodecagonallattices (chemical links) as function of p; and (b) of strips of different width at the percolationthreshold. deviation. The site percolation conductivity was studied only for the case of "chemical" links by means of the transfer-matrix approach to the random resistor network (RRN) problem. The R R N ' s was built supposing that two "open" lattice sites sharing a bond are connected by a unit resistor. Typical for the transfer-matrix approach is that long and relatively narrow strips from resistors are studied. The transverse conductivity of those strips should be determined for a set of different p and different widths. When p is fixed at p = Pc the scaling of the conductivity with the strip width allows one to estimate the percolation transport exponent for the model studied. The strips of resistors were 20000 bonds long and their width was varied up to 30 bonds. The conductivity scaling found for the both lattices is given on Fig. 3b. No significant deviation from the already known for the other 2D lattices critical behavior was found. The estimation error was around 5-10%: rather large for a reliable value for the percolation transport exponent. The conductivity of strips of fixed width (~-, 30) for p varied in the whole region of nonzero conductivities ( p ~ 0.5-1 ) was obtained as well. The results are given on Fig. 3a.
250
E Babalievski/Physica A 220 (1995) 245-250
In conclusion, the presented spanning curves for quasicrystalline lattices support the (limited) universality for percolation fractions at the percolation threshold in two dimensions. The percolation conductivity of dodecagonal and octagonal latices has the same behavior as the Penrose lattice and the periodic lattices. The estimates for the site percolation thresholds of the octagonal and dodecagonal lattice are given for two ways of linking the sites. These estimates could help in searching analytical expressions for the percolation thresholds. I thank V. Tonchev for useful discussions. This study was supported by the Bulgarian NSF Grant No. F-119/91.
References [ 1] [2] [3] [4] [5] [6] [7] [8] [9] 110] l 11 ] [ 12] [13] [ 14] [15]
H.Scher and R.Zallen, J. Chem. Phys. 53 (1970) 3759. M. Duneau and A. Katz, Phys. Rev. Lett. 54 (1985) 2688. U. Alon, I. Balberg and U. Drory, Phys. Rev. Lett. 66 (1991) 2879. J. P. Lu and J. L Birman, J. Stat. Phys. 40 (1987) 1057. J. Socolar, Phys. Rev. B 39 (1989) 10519. E Babalievski and O. Peshev, C. R. Acad. Sci. Bulg. 41 (1988) 85. E Yonesava, S. Sakamoto, K. Aoki, S. Nose and M. Hod, J. Non-Cryst. Solids 123 (1988) 73. D. Stauffer and A. Aharony, Introduction to percolation theory (Taylor and Francis, London, 1992); second printing 1994. D. Leduie and J. Teillet, unpublished. E Babalievski, unpublished. F. Babalievski and O. Peshev, Comput. Phys. Commun. 60 (1990) 27. B. Derrida, J.G. Zabolitzki, J. Vanimenus and D. Stauffer, J. Stat. Phys. 36 (1984) 31. E Babalievski, Z. Phys. B 84 (1991) 429. U. Gropengiesser and D. Stauffer, Physica A 210 (1994) 320, and references therein. C. Domb and M.E Sykes, Phys. Rev. 122 (1961)77.