DSpace at VNU: A continuum percolation model in an anisotropic medium: dimensional crossover?

By checking simultaneously two percolations, parallel and perpendicular to the chain direction, we show that while there is a quasi-one-dimensional to two-dimensional crossover in the pe

A continuum percolation model in an anisotropic

medium: dimensional crossover?

Nguyen Van Liena;∗, Dang Dinh Toib, Nguyen Hoai Namc

a Institute of Physics, P.O Box 429 Bo Ho, Hanoi 10000, Viet Nam

b Physics Faculty, Hanoi State University, 90 Nguyen Trai, Hanoi, Viet Nam

c Physics Faculty, Hanoi University of Education, Cau-giay, Hanoi, Viet Nam

Received 18 December 2001

Abstract

We propose a two-dimensional continuum percolation model in an anisotropic medium, which consists of parallel chains of sites coupled to each other weakly via rare “impurities” By checking simultaneously two percolations, parallel and perpendicular to the chain direction, we show that while there is a quasi-one-dimensional to two-dimensional crossover in the percolation radius

of 4nite systems as the “impurity” density s increases, in the limit of in4nite systems two percolations are equivalent in the sense that their main characters are, respectively, coincident, regardless of s The proposed model is assumed to be used for describing hopping conduction networks in the compounds such as conjugated polymers or porous silicon

c

 2002 Elsevier Science B.V All rights reserved.

PACS: 64.60.Ak; 64.60.Cn; 05.40.+j

Keywords: Two-dimensional percolation; Anisotropic medium; Dimensional crossover

1 The model

Recently, there has been a number of works reporting an observation of dimen-sional crossovers in the temperature dependence behaviour of the variable-range hop-ping (VRH) conductivity in various compounds, such as the conjugated polymers [1] or porous silicon [2] In di@erence from that induced by the electric 4eld [3], the crossovers reported in Refs [1,2] are believed to be associated with the speci4c

∗Corresponding author Tel.: +84-4-843-5917; fax: +84-4-8349050.

E-mail address: nvlien@iop.ncst.ac.vn (N Van Lien).

0378-4371/02/$ - see front matter c
2002 Elsevier Science B.V All rights reserved.

PII: S 0378-4371(02)01205-0

microstructure of materials It was shown that in conducting conjugated polymers far from the insulator-to-metal transition the conducting network consists of parallel linear chains which are weakly coupled to each other via rare impurities (or small metallic particles) At low doping levels (impurity concentration) [1] and=or low temperatures [2] the typical interchain hopping rate E is much less than the typical intrachain rate

I that gives rise to the quasi-one dimensional (Q1D) behaviour of conductivity With increasing doping level (or temperature) the interchain rate E increases rapidly and

at some critical doping level (temperature) it becomes comparable to the intrachain rate I, i.e., two directions, parallel and perpendicular to the chain direction, become equal in percolation This may result in a Q1D to 2D (3D) crossover in temperature dependence behaviour of VRH conductivity

It is widely accepted that the percolation method is one of the best approaches in describing the hopping conduction properties [4,5] To formulate a percolation model corresponding to the experimental systems of Refs [1,2] and to calculate its percolation characters in the 2D case are the aim of the present work

Let us consider the (L × L)-square with sides parallel to the x- and y-axis of the

Decartesian coordinates system Inside the square n linear chains of sites (atoms) are generated in the way that they are parallel with the x-direction and regularly sep-arated from each other Along each chain the sites are randomly arranged by the Poisson distribution with an average separation being equal to 1 (the length units in the simulation) Next, the impurities (small metallic islands) are modelled by sites added at random into the interchain spaces with a density s ¡ 1 In the result, we have, in general, a quasi-anisotropic 2D system of sites with random coordinates

as displayed schematically in Fig 1 Due to a randomness of site coordinates this system may be thought of as a continuum percolation model in a structurally anisotropic medium [5–7]

The model is thus characterized by two physical parameters: the “chain density”

a chain is equal to 1) In the limit of n = 1 and s = 0, we have a 4nite 1D system

of length L In the other limit of s → 1 the system should tend to behave like a

is generally anisotropic in the sense that in the direction longitudinal to the chains the percolation proceeds easier than that in the perpendicular direction In other words, for the same 4nite sample with a low density s, the threshold radius R(t)c of percolating from the upper to the lower edge of the square sample (see, Fig 1) in the direction perpendicular to the chains (transverse percolation—TP) should be greater than the corresponding radius Rc(l) of percolating from the left to the right edge along the chain direction (longitudinal percolation—LP): R(t)c =R(l)c ¿ 1 Certainly, with increasing the density s, the ratio R(t)c =R(l)c decreases and it is natural to assume that there must exist

a critical density s = sc such that for all s ¿ sc two percolation radii are practically coincident Such a change in the relation between two percolation radii could perhaps produce a dimensional crossover in the temperature dependence behaviour of the VRH conductivity since the latter is mainly determined by the typical hopping length, which

is in turn associated with the corresponding percolation radius [5] De4nitely, the critical density sc depends on the chain density However, even for a given , it is still not

Fig 1 The simulation model: medium is anisotropic, but percolating process is isotropic (circle problem).

clear (i) how the crossover behaviour depends on the sample size L (if there has a crossover in an in4nite system) and (ii) if other percolation characters, namely, the percolation exponent  [4,5] and the fractal dimension D [4], experience the same crossover As an attempt to 4nd answers to these questions, in the present work both percolations, TP and LP, are simultaneously simulated in samples with di@erent sizes

L and di@erent values of s for some typical values of the chain density The assumed dimensional crossover will be checked by comparing corresponding characters of LP and TP, taking into account the 4nite size e@ects

2 Calculations and numerical results

random sites from the left edge towards the right edge of the simulation square (i.e., along the chain direction (LP) in Fig 1) and from the upper edge towards the lower one (TP) As usual [4,5], regarding a particularity of the edge to edge percolation problem, the periodical boundary condition is only applied in the direction perpendicular to the percolating direction

For each direction the threshold radius can be evaluated using the standard algorithm for the so-called r-percolation problem [4,5] Given a distance R, any two sites i and j with distance rij are assumed connected by a bond if rij6 R or disconnected if rij¿ R The threshold percolation radius is then de4ned as the lowest R such that the bonds

rij6 R form a percolating network connecting two edges Such a percolating network

is often called the critical cluster (the largest cluster of connected sites at percolation) and the number of sites belonging to it is called the cluster mass (see later, Fig 5 as

an illustration of critical clusters)

Thus, for each realization of site coordinates we can calculate the percolation radii

R(l)c (L) and R(t)c (L), as well as, the critical cluster masses M(l)(L) and M(t)(L) for

LP and TP, respectively In general, the quantities R(l);(t)c (L) and M(l);(t)(L) Luctu-ate from one random site realization to another Their averages over site realizations will de4ne the percolation radii and the critical masses for samples of given size L:

R(l);(t)c (L) = R(l);(t)c (L) and M(l);(t)(L) = M(l);(t)(L), respectively The fact that both

quantities R(l);(t)c (L) and M(l);(t)(L) depend on the sample size L is a natural conse-quence of the 4nite size e@ects of the investigated problem In order to study these e@ects and to 4nd the percolation radii, as well as other percolation characters, corre-sponding to an in4nite system, for given values of and s the simulations have been performed for samples of di@erent sizes: L = 100, 200, 400, 600, 800 and 1000 The number of random site realizations over which the averages are taken is k = 5000 for

the case L = 100 and decreases as L increases in such a way that the number k × L2, i.e., the total “area” used in checking percolation, are almost constant for all the cases under study

In simulations, along with the average quantities R(l);(t)c (L) and M(l);(t)(L) we always calculate the moments [4,5]:

R(l);(t)

c (L) = (R(l);(t)

c (L) − R(l);(t)

M(l);(t)(L) = (M(l);(t)(L) − M(l);(t)(L))21=2; (2) and also the next-order moments, which are used to estimate the statistical errors of simulation results

The quantities R(l);(t)c , which measure the width of transition regions of the span-ning clusters and depend on the sample size L, can be used to de4ne the percolation exponents (l);(t), following the well-known 4nite-size scaling relation [4,5,8]

R(l);(t)

In Fig 2, the simulation data of −ln R(l);(t)c (L) are plotted versus ln L for the case =0:1 and some values of the impurity density s, s=0:05, 0.1, 0.2, 0.3, and 0.4 (from bottom) For each value of s the simulation points are described by the solid circles and the triangles for the LP and TP, respectively Everywhere in this and following 4gures the error bars do not exceed the symbol sizes It is clear from Fig 2 that for each s the simulation points for both LP and TP follow quite well the scaling relation of

Eq (3) More importantly, the slopes of all 4tting straight lines, solid lines for LP and

dashed lines for TP, fall well into a narrow value range: 0:71±0:02, which corresponds

to  = 1:41 ± 0:04 We note that what observed for some values of s in Fig.2 is also

maintained for unshown cases with other values of s, 0 ¡ s ¡ 1 Therefore, within the statistical error (less than 3%), our simulation data suggest that for the model under study with a given there exists only one critical exponent  for both LP and TP and for any s of 0 ¡ s ¡ 1

4 5 6 7 1

2 3

lnL

Fig 2 −ln R(l)c (L)(•) and −ln Rc(t)(L)() are plotted versus ln L for the cases of = 0:1 with di@erent

s (from bottom): s = 0:05, 0.1, 0.2, 0.3, 0.4.

Next, for each studied case the percolation radii rc(l);(t)(s) ≡ R(l);(t)c (L → ∞),

corre-sponding to an in4nite system, can be determined using the scaling relation [4,5,8]:

|r(l);(t)

c − R(l);(t)

In Fig 3, the quantities R(l);(t)c (L) are plotted against L−1=(l); (t) for the case of = 0:1 and di@erent values of s: s = 0:01, 0.05, 0.1, 0.2, 0.3, and 0.4 (from top), using (l);(t) determined directly from Fig 2 It is clear that for any 4nite sample size L, for all values of s under study the TP radius R(t)c (L) (triangles) is always higher than the

LP one R(l)c (L) (solid circles) However, with increasing L two percolation radii seem

to change by di@erent directions: R(l)c (L) increases, while R(t)c (L) decreases Such a di@erence on direction of the 4nite size e@ects associated with LP and TP is similar

to that discussed in Ref [6] The 4nite size e@ect in the LP, as thoroughly discussed

in Ref [9], is mostly associated with the sites located close to the boundaries For the

TP, the e@ect can be understood by the fact that if the distance between two adjacent parallel chains is 4xed (i.e., given ), then the longer the chain length L is, the shorter the optimum percolation path between them becomes The fact that the 4nite size e@ects for the LP and TP are in opposite directions is a remarkable characteristic of the studied percolation model On the other side, importantly, as is evident in Fig 3, for any density s under study and for both the LP and TP the simulation points obey well the scaling relation of Eq (4) And therefore, the asymptotics of the 4tting straight lines (solid or dashed for LP or TP, respectively) give the corresponding percolation radii (rc(l) or rc(t)) of in4nite systems

Very impressively, although for any 4nite L the radius Rc(t)(L) is always considerably larger than R(l)c (L) (the smaller s, the larger discrepancy), due to a di@erence on the direction of 4nite size e@ects as mentioned above for each s under study in the limit

of L → ∞ two radii tend to the practically coincided limits: rc(l)(s) = rc(t)(s) ≡ rc(s)

0 0.02 0.04 0.06 1

3 5 7

Fig 3 The 4nite size e@ect of percolation radii R(l);(t)c (L) are plotted against L−1=(l);(t) (•and  for LP

(l) and TP (t), respectively) for the cases of = 0:1 with di@erent s (from top): s = 0:01, 0.05, 0.1, 0.2, 0.3, 0.4 The extrapolations of 4tting straight lines yield the values of r c(l);(t), corresponding to in4nite systems.

The radius rc(s) decreases with increasing s and with a relative error less than 3% Fig 3 gives rc(s) ≈ 4:987, 3.534, 2.898, 2.271, 1.943, and 1.726 for s = 0:01, 0.05,

0.1, 0.2, 0.3, and 0.4, respectively Thus, our simulation results suggest that while there

is a di@erence between percolation radii of the LP and TP in 4nite systems, for any impurity density 0 ¡ s ¡ 1, in the limit of in4nite systems two correspondingly limiting radii are coincident

One more important character of the percolation problem under study as mentioned above is the scaling behaviour of the critical cluster mass, i.e., the feature of the average

total number of sites M(L) ≡ M(L) belonging to the largest percolative cluster at threshold in dependence on the sample size L as L → ∞ It was originally suggested by

Mandelbrot [10] and is now widely believed that the critical cluster exhibits a fractal structure and its mass scales with L as

where D is called the fractal dimension

4 5 6 7 4

6 8 10

lnL

Fig 4 ln M (l) (•) and ln M (t)() are plotted against ln L for the cases of = 0:1 with di@erent s (from

top): s = 0:05, 0.1, 0.2, 0.3, 0.4 The slopes of 4tting straight lines give the fractal dimension D.

In the present simulation for given L, , and s the mass M was counted for both

LP (M(l)) and TP (M(t)) In Fig 4, we plot ln M(l)(L) (solid circles) and ln M(t)(L) (triangles) against ln L for the case = 0:1 and some values of the impurity den-sity: s = 0:05, 0.1, 0.2, 0.3, and 0.4 (from top) The straight lines (solid or dashed for LP or TP, respectively) are the best 4ts of simulation points to the relation of

Eq (5) It is clear that for all the cases under study the simulation data follow well the Mandelbrot’s scaling relation of Eq (5) And very interestingly, the fractal dimen-sions D as measured by the slopes of the 4tting lines are practically the same for

both LP and TP and for all s under study From Fig 4 we obtain D = 1:94 ± 0:04.

We like here to note that for any sample of 4nite L, as can be seen in Fig 4, the mass M(t)(L) of TP is always larger than the mass M(l)(L) of LP This is even more evident when we compare the cluster in Fig 5a (for LP) with that in Fig 5b (for TP)

or the cluster in Fig 5c (for LP) with that in Fig 5d (for TP) for s = 0:1 or 0.3, respectively However, importantly, for all cases under study both these masses scale with the sample size L with the same value of the power D in Eq (5)

Fig 5 Examples of critical clusters for sample of L = 400 and = 0:1 (a) and (c): LP for s = 0:1 and 0.3; (b) and (d): TP for s = 0:1 and 0.3, respectively.

Thus, with respect to all three characters (percolation exponent, percolation radius, and fractal dimension) our simulation results in Figs 2–4 for the case = 0:1 show that for in4nite samples two percolation, LP and TP, are equivalent, i.e., the system could be considered isotropically 2D percolative, regardless of s

The similar calculations have also been performed for some other values of the chain density in the range 0:01 6 6 0:2 (in real systems the distance between adjacent chains is always much larger than the average distance between nearest neighbouring sites in a chain [11]) The obtained results are qualitatively similar to those for the case of = 0:1, presented in Figs 2–4 They all suggest an isotropic 2D behaviour

of percolation in in4nite systems with a single  as well as D for both LP and TP, regardless of s And, very surprisingly, within statistical errors the values of these characters seem to be also insensitive to a change of in the range of values under study although the limiting percolation radii rc(l);(t)(L → ∞) and the 4nite-size critical

density sc(L) clearly depend on (see the next section, Fig 6) Thus, while the

0 0.004 0.008 0.012 0

0.2 0.4 0.6 0.8

L−1

s c

Fig 6 The size-dependent “critical” densities s c (L) for 4nite systems are plotted versus L−1for some values

of (from top): 0.05, 0.1, 0.2 The curves are freely drawn as a guide.

percolation radii depend on both and s, the percolation exponent  ≈ 1:41 ± 0:04 and the fractal dimension D ≈ 1:94 ± 0:04 may be considered universal at least for the

ranges of parameters under study

3 Discussion

Our simulation results (Figs 2–4 and unshown for other values of and s) suggest that in the investigated model (1) there have a single critical exponent and a single fractal dimension which are independent of percolation directions (LP or TP), of the impurity density s with 0 ¡ s ¡ 1, and of in the range of most practical interest 0:01 6 6 0:2 and (2) though for 4nite systems the threshold radius of the transverse

percolation R(t)c (L) is always larger than that of the longitudinal percolation R(l)c (L), for in4nite systems two radii are coincident and the single radius rc ≡ Rc(t)(L →

given and s in the ranges of values under study two percolations, LP and TP, are equivalent in the sense that all their characters, , D and threshold radius, are, respectively, coincident In other words, the system becomes then isotropically 2D percolative, regardless of an anisotropicity in its structure

In order to understand the obtained results we assume that, generally, in in4nite systems an anisotropicity in percolation is not associated with the medium topological structure, but rather with the rule of generating percolation clusters1 (an anisotropicity

of medium should be averaged out as the system size extends to in4nity) In a 2D isotropic and homogeneous medium, for example, two di@erent rules, namely, the circle (isotropic) rule and the forward (directed) one, as well known, lead to di@erent classes

of percolation, characterized by di@erent values of  and D: for isotropic percolation

 = 0= 4=3 and D = D0= 91=48 (exact values [4]); for directed percolation  ≈ 1:73,

rules are isotropic (circle) and identical for both LP and TP, therefore two percolations (LP and TP) should belong to the same class of the same  and the same D The nature (e.g., the topological structure) of the system should be manifested in the values of

these quantities:  ≈ 1:41 and D ≈ 1:94 We have no idea to explain quantitatively the

di@erences between the values of  and D obtained in the present model and those of

0 and D0 mentioned above, but only assuming that they may be related to a di@erence

on topological structure of two corresponding systems Besides, we like to note that the obtained values of  and D are not in contrast to the well-known percolation relation

D = d − =, where d = 2 and  = 5=36 [13].

Concerning the percolation radius rc(s) of in4nite systems we have some guesses for the limiting cases When s is close to 1, the system could be seen quasi-isotropic, and therefore, rc(s) should be close to the value 1.19, determined from the percolation relation r2= Bc with Bc= 4:5 for the circle problem [5] In the opposite limit, when

s → 0, the radius rc will be bounded by the distance between adjacent chains, which

is equal to 1= (i.e., 10 for data in Fig 3) Thus, we have 1:19 6 rc(s) 6 −1, which was really observed in simulation data (particularly, in Fig 3 for = 0:1)

Now, while there is no crossover on percolation radius for in4nite systems, one can ask if there is any quasi-crossover in 4nite systems To 4nd an answer to this question, with a given for each sample size L of 100 6 L 6 1000 we have searched if there

is a crossover on the percolation radius as the density s varies Note again that the hopping conductivity is mainly determined by the percolation radius For each L our simulations show that there really exists a size-dependent “critical” density sc(L) such that R(t)c (L) ¿ R(l)c (L) for all s ¡ sc(L), while for all s ¿ sc(L) two radii are coinci-dent Certainly, the density sc(L) also depends on For a given , as can be seen in Fig 6, sc(L) strongly decreases with increasing L The fact that sc(L) monotonously decreases as L increases unambiguously implies that in the limit of in4nite L two

1 This reminds us of the generating rule in constructing fractals: each rule leads to a class of fractals with

a well-de4ned D (see Refs [10,13]).

In order to understand the obtained results we assume that, generally, in in4nite systems an anisotropicity in percolation is not associated with... (L) and M(l);(t)(L) Luctu-ate from one random site realization to another Their averages over site realizations will de4ne the percolation radii and the critical masses for samples... class="page_container" data-page="10">

percolation R(t)c (L) is always larger than that of the longitudinal percolation R(l)c (L), for in4 nite