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Percolation processes in monomer-polyatomic mixtures
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Physica A 374 (2007) 239–250 www.elsevier.com/locate/physa
Percolation processes in monomer-polyatomic mixtures M. Dolza, F. Nietob, A.J. Ramirez-Pastorb, a
Centro Ato´mico Bariloche, CONICET, Av. Bustillo 9500, 8400, S. C. de Bariloche, Argentina Departamento de Fı´sica, Universidad Nacional de San Luis, CONICET, Chacabuco 917, D5700BWS San Luis, Argentina
b
Received 13 September 2005; received in revised form 22 May 2006 Available online 21 July 2006
Abstract In this paper, the percolation of mixtures of monomers and polyatomic species (k-mers) on a square lattice is studied. By means of a finite-size scaling analysis, the critical exponents and the scaling collapsing of the fraction of percolating lattices are found. A phase diagram separating a percolating from a non-percolating region is determined. The main features of the phase diagram are discussed in order to predict its evolution for larger k-mer sizes. r 2006 Elsevier B.V. All rights reserved. Keywords: Percolation; Monte Carlo simulations; Finite-size scaling theory; Phase transitions
1. Introduction The study of clustering and percolation behavior in different physical systems is related to a wide range of phenomena of both theoretical and technological importance [1–10]. Many of these real world systems are best modeled by percolation of randomly distributed objects of a given shape and size as opposite to standard site or bond percolation in which single sites or bonds in a discrete lattice are randomly occupied. While certain quantities, i.e. critical exponents, are independent of the details of the percolation model other ones such as the percolation threshold are not [11–17]. However, most of the studies are devoted to the percolation of molecules with single occupancy. Thus, whether some sort of correlation exists, like particles occupying several ðkÞ contiguous lattice sites (k-mers), the statistical problem (multisite statistics) becomes exceedingly difficult. In this line of thinking, several authors have produced seminal contributions for analyzing the percolation of polyatomic species on different lattices [18–24]. More recently [25–27], the percolation of (a) linear segments of size k and (b) k-mers of different structures and forms deposited on a square lattice have been studied. In addition, a generalization of the pure site and pure bond percolation problems in which pairs of nearest neighbor sites (site dimers) and linear pairs of nearest neighbor bonds (bond dimers) are independently occupied at random on a square lattice was discussed in order to determine the phase diagram of the system. However, the problem is far to be exhausted. In this context, the present paper deals with the percolation of Corresponding author. Tel.: +54 2652 436151; fax: +54 2652 430224.
E-mail addresses: [email protected] (M. Dolz), [email protected] (F. Nieto), [email protected] (A.J. Ramirez-Pastor). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.06.017
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mixtures of two kind of species, namely, monomers and k-mers ðk41Þ on a regular two-dimensional lattice. Physically, this could correspond to irreversible co-adsorption of several different molecular species or, alternatively, adsorption of a single type of molecule with different binding configurations. The latter is proposed for CO adsorption on several metal surfaces. This process involves competition between two-point binding b CO at pairs of sites and one-point binding of a CO at single sites [28]. From a theoretical point of view, the competitive irreversible adsorption (or random sequential adsorption, RSA) of mixtures of monomers and k-mers has been widely studied in the literature [23,29–31]. In fact, Evans and Nord used highorder hierarchical truncation to analyze competitive filling of various small animals on a square lattice [18]. Barker and Grimson simulated RSA of mixtures of 4-mers of different shapes [30]. On the other hand, kinetics of RSA on a square lattice for mixtures of line segments of two different lengths has been studied by Svrakic and Henkel [23,31]. Other examples are compiled in Ref. [32]. A study as presented here is a natural extension of those papers, where the distributions of empty and occupied sites generated by RSA are analyzed in the framework of the percolation theory. Based in a Monte Carlo (MC) analysis, the main aim of the paper is (a) to include dynamic in the model by establishing a parameter which take into account the probability of deposition for each kind of percolating species; (b) to consider the evolution of the percolation threshold under the variation of both the relative concentration of percolation species and the size of them and (c) to determine the phase diagram as a function of the parameters of the problem in order to explain the percolative properties of the system. The paper is organized as follows. The basic definitions related to the model and the numerical techniques used in the MC analysis are presented in Section 2. In fact, in Section 2.1 the basis of the model of deposition of monomer-polyatomic mixtures on a site square lattice is given. Section 2.2 is devoted to describe the numerical method used throughout the paper, including finite-size scaling theory. The characteristic phase diagrams, along with the collapsing curves predicted by the finite-size scaling theory are shown in Section 3. Finally, conclusions are drawn in Section 4. 2. Basic definitions and details of the computer simulation 2.1. The model Let us consider a square lattice of linear size L with periodic boundary conditions. Monomers and k-mers are deposited onto such lattice until partial concentrations ð1 cÞp and cp of such particles are reached, respectively. Accordingly, (i) p represents the fraction of occupied sites, and (ii) c (ranging between 0 . . . 1) is defined as the relative concentration of k-mers that will be present in the mixture. In other words, c is the number of sites occupied by k-mers over the total number of occupied sites. The fact that one component of the mixture occupies more than one site requires the definition of a dynamic for the irreversible deposition.1 For this purpose, we include an extra parameter ðtÞ which is defined in the framework of the following scheme. First, a random number, x is selected. If x is larger than a given probability t a monomer is candidate to be deposited. Then, if both (a) the concentration of monomers is lower than ð1 cÞ and (b) a site randomly selected is vacant; the site becomes occupied. Otherwise the attempt is rejected. For values of x lower than t a k-mer is considered to be deposited whether the relative concentration of polyatomics is lower than c. In this case, a k-uple of nearest neighbor sites is randomly selected; if it is vacant, the k-mer is then dropped on those sites. Otherwise, the attempt is rejected. Thus, if t ¼ 0 the ð1 cÞpL2 monomers will be deposited first and then the L2 cp=k k-mers are dropped onto the lattice while in the opposite limit whether t ¼ 1 will be the cpL2 =k k-mers first deposited and next the corresponding ð1 cÞpL2 12 monomers. In Fig. 1 a lattice of L ¼ 6, p ¼ 19 36, c ¼ 19 and t ¼ 1 is presented for illustrative purposes where monomers (hexagons in the figure) and trimers (circles) are deposited. The central idea of the percolation theory is based in finding the minimum concentration p for which at least a cluster (a group of occupied sites in such a way that each site has at least one occupied nearest neighbor site) extends from one side to the opposite one of the system. This particular value of the concentration rate is 1
As an example, note that whether cp ½ð1 cÞp k-mers [monomers] are deposited first, the saturation limit p ¼ 1 can [cannot] be reached.
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k=1
k=3
Fig. 1. An illustrative example of the model for a square lattice of sites with size L ¼ 6. Monomers (hexagons) and trimers (circles) are 12 dropped onto the lattice with the following values of the parameters: p ¼ 19 36, c ¼ 19 and t ¼ 1.
named critical concentration or percolation threshold and determines a phase transition in the system. In the random percolation model, a single site is occupied with probability p. For the precise value of pc , the percolation threshold of sites, at least one spanning cluster connects the borders of the system [indeed, there exist a finite probability of finding n ð41Þ spanning clusters [33–36]]. In that case, a second-order phase transition appears at pc which is characterized by well defined critical exponents. As it was already mentioned, the main goal of this paper is (a) to determine how the mixture of different kinds of k-mers influences in the numerical value of the percolation threshold of the system and (b) to verify the universality class of the phase transition involved in the problem. For these purposes, long scale numerical simulations (independent of the size of the k-mer) are required in order to predict the behavior of the system in the thermodynamic limit. The ratio L=k is kept constant for avoiding spurious effects due to the k-mer size in comparison with the lattice size. A study of the finite-size effects allows us to make a reliable extrapolation to the k ! 1 limit when the limit L ! 1 is taken before. Details of this study will be given in the next subsection. 2.2. Monte Carlo simulations It is well known that it is a quite difficult matter to analytically determine the value of the percolation threshold for a given lattice [2,4,5,7,8]. For some special types of lattices, geometrical considerations enable to derive their percolation thresholds exactly. For systems which do not present such a topological advantage, percolation thresholds have to be estimated numerically by means of computer simulations. Percolation of polyatomic species adds a new ingredient to the problem: the influence of local correlation. As the scaling theory predicts [9], the larger the system size to study, the more accurate the values of the threshold obtained therefrom. Thus, the finite-size scaling theory gives us the basis to achieve the percolation threshold and the critical exponents of a system with a reasonable accuracy. For this purpose, the probability R ¼ RX L ðpÞ that a lattice composed of L L sites percolates at concentration p can be defined [2]. Here, as in Ref. [37,38], the following definitions can be given according to the meaning of X : (a) RIL ðpÞ ¼ the probability that we find a cluster which percolates both in a rightward and in a downward direction; (b) RU L ðpÞ ¼ the R 1 probability of finding either a rightward or a downward percolating cluster and (c) RA L ðpÞ 2 ½RL ðpÞ þ D I U 1 RL ðpÞ 2 ½RL ðpÞ þ RL ðpÞ. In order to determine the percolation threshold, it is necessary to evaluate the effective threshold pc ðLÞ for a lattice of finite size L. In the MC simulations, RX L ðpÞ is calculated for each discrete value of p according to the considered regular finite lattice. For determining the desired quantities two steps are required: (a) the construction of m samples for a given coverage (according to the scheme presented in Section 2.1) and (b) the
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cluster analysis by using the Hoshen and Kopelman algorithm [39]. In the last step, it is verified whether exists a percolating island. This spanning cluster could be determined by using the criteria I, U or A. n runs of such two steps are carried out for obtaining the number mX of them for which a percolating cluster of the desired X criterion X is found. Then, RX L ðpÞ ¼ m =n is defined and the procedure is repeated for different values of the parameters of the problem. A In Fig. 2a (b), the probabilities RIL ðpÞ (squares), RU L ðpÞ (circles) and RL ðpÞ (triangles) are presented for the problem of site percolation of monomers and dimers (trimers) at two different concentration as indicated and probability t ¼ 0. From a first inspection of the figure (and from data do not shown here for a sake of clarity) several conclusions are in order. Curves cross each other in a unique universal point, RX , which depends on the criterion X used. The abscissa of this point is pc ð1Þ. Those points do not modify their numerical value for the different both mixtures of polyatomic species used and values of the parameters c and t. R is also called in the literature as percolation cumulant, whose properties are identical to those of the Binder cumulant in standard thermal transitions [9,40]. Here, we have discussed the behavior of this cumulant obtained via three different criteria. One of the most important characteristics of RX shows that its value at the critical threshold is a universal quantity, i.e. is the same for models in the same universality class [41,42]. The observed findings
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0.58 pc(∞)
A
0.57 I 0.56 0.00
(c)
0.02
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Fig. 2. (a) Fraction of percolating lattices as a function p. Different criteria are used to identify the spanning cluster, namely, RU L ðpÞ of the (circles); RIL ðpÞ (squares) and RA L ðpÞ (triangles). This example corresponds to the mixture monomer–dimer and the particular values parameters c ¼ 0:95 (full symbols) and c ¼ 0:20 (empty symbols) and t ¼ 0 for both cases. Horizontal dashed lines show the RX universal points. Vertical dashed lines denote the percolation threshold, pc in the thermodynamic limit L ! 1. (b) As Fig. 2(a) for the mixture monomer–trimer and the parameters of the problem as indicated. (c) Extrapolation of pc ðLÞ of Fig. 2a towards the thermodynamic limit according to the theoretical prediction given by Eq. (1). Squares, triangles and circles denote the values of pc ðLÞ obtained by using the criteria I, A and U, respectively. The error bars are smaller that the symbol size.
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are then a clear indication (as it is expected) that the problem belongs to the same universality class regardless both the mixture considered and the values of the parameter of the model. X For each curve RX L ðpÞ, pc ðLÞ can be obtained from the value in which the slope is maximum. The X extrapolation of pc ðLÞ toward the limit L ! 1 is predicted to follow: X 1=n , pX c ðLÞ ¼ pc ð1Þ þ A L
(1)
X
where A is a non-universal constant and n is the critical exponent associated with the correlation length which is analytically shown to be equal to n ¼ 43 in the case of random percolation. Fig. 2c shows the extrapolation towards the thermodynamic limit of pX c ðLÞ according to Eq. (1) for the parameters used in Fig. 2a corresponding to c ¼ 0:95. This figure lends support to the assertion given by Eq. (1): (a) all the curves (different criteria) are well correlated by a linear function, (b) they have a quite similar value for the ordinate in the limit L ! 1 and (c) the fitting determines a different value of the constant A depending of the type of criterion used. It is also important to note that pA c ðLÞ gives a perfect horizontal line which is a great advantage of the method because it does not require precise values of critical exponents in the process of estimating U A percolation thresholds. The maximum of the differences between jpIc ð1Þ pA c ð1Þj and jpc ð1Þ pc ð1Þj gives the error bar for each determination of pc . For technical details in the application of the method, (in particular, for determining the critical exponents) we refer the interested reader to Refs. [25–27,37,38]. 3. Results and discussions 3.1. Phase diagrams We focus now on the influence of the filling dynamics on the percolation threshold upon varying the probability t from 0 to 1 while c remains constant. These curves are shown in Fig. 3 for a monomer-dimer mixture and different values of c, as indicated. Several considerations are in order about this behavior. In particular, two different regimes can be distinguished. The limit t ¼ 0 reflects the situation where all the particles dropped first are monomers. Then, the k-mers are deposited in the vacant sites. Upon increasing smoothly t a increasing fraction of k-mers are deposited when not all the monomers are dropped. The local correlation introduced by the k-mers produce more compact structures which in turn slightly reduce the percolation threshold. In the opposite limit, for t ¼ 1 the interstitial holes needed to be filled for building the spanning cluster must be occupied by monomers. This situation is favorable as compared with the case t ¼ 0 where k-mers must fill the vacant sites and a larger value of surface coverage is needed for connecting the extremes of the lattice being always pc ðt ¼ 0Þ4pc ðt ¼ 1Þ. The above physical considerations constitute two
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0.565 0.0
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Fig. 3. Percolation thresholds as a function of t for several values of the constant c, as indicated. The curves shown such behavior for a monomer–dimer mixture.
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different regimes for the limit cases t ¼ 0 and t ¼ 1. The curves show a inflexion point between these two different regimes approximately when t c. The numerical value of pc ðtÞ remains almost constant far away of t c, i.e. ( pc ð0Þ if t5c pc ðtÞ (2) pc ð1Þ if tbc: Data from Fig. 3 permit to build Fig. 4 where the percolation thresholds are now plotted as a function of c being the curves parametrized by t. Three regions can be distinguished. The first one corresponds to the nonpercolating (NP) area characterized by the relationship pðc; tÞopc ðc; 1Þ. In a second region where pðc; tÞ4pc ðc; 0Þ, the system always percolates and is labeled as P1 in the figure. Finally, the dashed area P2 corresponds to surface concentrations which percolate depending of the value of t, for each c, according to Fig. 3. pc ðc; t ¼ 1Þ is a monotonically decreasing curve. Upon increasing the concentration of k-mers in the lattice the percolation threshold decreases as a consequence of the local correlation in agreement with Refs. [25,26,43,44]. However, the curve pc ðc; t ¼ 0Þ increases as c is raised, then goes through a definite maximum and finally decreases for larger values of c. This behavior can be understood as follows. For c ¼ 0 (all the particles on the lattice are monomers) the percolation threshold is the well-known value pc ¼ 0:5927 [2]. In this conditions, the minimum increasing of c corresponds to an exchange of one monomer by one dimer. This procedure clearly deals to a new critical concentration larger than the percolation threshold for the case of monomers ðc ¼ 0Þ. Obviously, this effect is enhanced for larger values of k. However, as c increases, the local correlation present due to the adsorption of k-mers limits the above described tendency. Then, the curve reaches a defined maximum and monotonically diminishes down to pc ð1; 0Þ. This extreme value pc ðc ¼ 1; t ¼ 0Þ ¼ pc ðc ¼ 1; t ¼ 1Þ is the percolation threshold of dimers on a square lattice [25,26]. Figs. 3 and 4 represent the case of monomer-dimer mixtures while Fig. 5 shows the evolution of those curves for different monomerk-mer mixtures, as indicated. For all the curves considered, the initial point is the same pc ðc ¼ 0; tÞ ¼ 0:5927. In the other extreme, c ¼ 1 the percolation threshold is only a function of k which is given by pc ðc ¼ 1; t ¼ 0Þ ¼ pc ðc ¼ 1; t ¼ 1Þ ¼ pc ðkÞ. The numerical values of such function have been obtained in Refs. [25,26] where it was shown that pc ðk ! 1Þ 0:461. In order to predict the evolution of these curves in the thermodynamic limit k ! 1, we consider the relevant features of them. The area p, defined as: p ¼ \P2 ,
(3)
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P1 0.58
NP 0.57
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Fig. 4. Percolation thresholds are plotted as a function of c while t remains constant. Three regions can be distinguished: (a) the nonpercolating (NP) area characterized by the relationship pðc; tÞopc ðc; 1Þ; (b) P1 , where the system always percolates, pðc; tÞ4pc ðc; 0Þ and (c) the dashed area P2 , where the system percolates depending of the value of t for each c. The curves show the behavior for a monomer–dimer mixture.
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Fig. 5. Idem Fig. 4 for different mixtures, as indicated. The quantity p, defined through Eq. (3), is represented by the dashed area.
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Fig. 6. (a) Evolution of the area p as function of the k-mer size. (b) Difference Dpc pc ðc; t ¼ 0Þ pc ðc; t ¼ 1Þ versus c for several monomer-k-mers mixtures.
is represented as a dashed region in Fig. 5 and its evolution is shown in Fig. 6a as a function of the k-mers size. The dots fit very well with a decreasing exponential function whose limit value is pðk ! 1Þ 0:0017. On the other hand, the difference Dpc pc ðc; t ¼ 0Þ pc ðc; t ¼ 1Þ is plotted as a function of c for several monomer-kmers mixtures, as indicated in Fig. 6b. It is observed that Dpc tends to a limiting curve as the size of the k-mers
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increases. By considering both features, the curve with spheres in Fig. 7 is postulated as the response of the system in the thermodynamic limit k ! 1. Finally, a limiting curve separating the percolating and the NP region independently of both: (a) the filling mechanism (in other words, regardless the value of t) and; (b) the k-mer size can be defined. Thus, for each value of c, the largest value of the percolation threshold, pc ðc; t ¼ 0; kÞ, is collected in a single curve. The result of the procedure is shown in Fig. 8. 3.2. Finite-size scaling analysis In this subsection, the influence of the parameters c and k on the collapsing curves is studied in the light of the finite-size scaling theory. The scaling law hypothesis predicts the collapsing of the curves RX L ðpÞ when they
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C Fig. 7. Idem Fig. 4 for different mixtures. The prediction of the limiting curve for k ! 1 is included. p1 represent the tendency of p for k ! 1 according to Fig. 6b.
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Fig. 8. The limiting curves separating the percolating and the non-percolating regions independently of both (a) the filling mechanism and (b) the k-mer size. Indicated portions of the curve correspond to different monomer-k-mer mixtures.
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are plotted as a function of a reduced variable u ðp pc ÞL1=n : 1=n X RX , L ðpÞ ¼ R ½ðp pc ÞL
(4)
being RX ðuÞ the scaling function. Thus, RX is an universal function with respect to the variable u. In Fig. 9, as an illustration, we plot RA L as a function of u for (a) monomer–dimer; (b) monomer–trimer and (c) monomer–tetramer mixtures and several values of concentration as indicated. The collapse gives an additional indication for the numerical value of the critical exponent n. As it is clearly seen from this analysis, the problem belongs to the same universality class of random percolation regardless of the value of c and k considered. In all cases, the curves nicely collapse into a unique universal curve according to the theoretical prediction. However, the scaling function seems to be different for each mixture considered. In Fig. 10, a representative curve of each mixture has been plotted as a function of u. This fact determines that the scaling function RX is not an universal function with respect to the variable k (each value of k is represented by using a different symbol, as indicated). 1.0 (a) 0.10
RAL (P)
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1=n Fig. 9. RA for (a) monomer–dimer; (b) monomer–trimer and (c) monomer–tetramer mixtures and several L as a function of u ðp pc ÞL values of concentration c.
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Fig. 10. Representative curves of several mixtures plotted as a function of u.
In order to determine the dependence of RX with k, the main features of the collapsing data have been considered in the range of k between 2 and 7. The curves become steeper upon increasing the value of k. Thus, we can observe that: (a) The derivatives become more pronounced as k increases. It is possible to establish a power-law to describe this behavior, ! qRX ¼ Bkr . (5) qu max
In a log–log scale the points are very well correlated by a linear function, like in Eq. (5), being the numerical values of the fitting parameters very similar for the three criteria used here. (b) The derivatives are narrowed upon increasing k. This behavior can also be described by a power-law according to: DX ¼ Ckl , X
(6) X
where D is the standard deviation of ðqR =quÞ for each curve. Thus, the standard deviation of each derivative versus k, when is plotted in a log–log scale, is very well correlated by a linear function, being the fitting parameter l ¼ 0:95 0:02 for A, U and I criteria (not shown here). According to the equations above, a metric factor might be included in the scaling function, Eq. (4), in order to collapse all the curves in Fig. 10 onto a single one. Following Ref. [40], in Fig. 11 we plot the probability RX L as a function of the argument z ¼ ðp pc ÞL1=n kl . The curves nicely collapse in the close vicinity of z ¼ 0 (close to the critical point) and a tiny deviation is observed as jzj increases. The scaling analysis given above should be rigorously valid only for sufficiently large L and from p in the asymptotic critical regime. However, it can be applied to the entire range of L and p if the data falls in the ‘‘domain of attraction’’ of a simple fixed point characterizing only one universality class of critical phenomena [9]. It is remarkable that more than 3 103 points are included in the collapsing curve. The metric factor introduced here, kl , gives an additional indication for the numerical value of the exponent l obtained in Eq. (6). 4. Conclusions In this paper, we have studied the percolation process for monomer-k-mer mixtures with increasing values of k. The percolating species, monomers and k-mers, are deposited at different relative concentrations, (1 c)
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5
(p-pc)L1/v k 1=n l Fig. 11. RA k , where l is obtained from Eq. (6). L as a function of the argument z ¼ ðp pc ÞL
and c, respectively. In addition, we have taken into account different probabilities of deposition for each species. This study does not attempt to reproduce any particular experimental situation but it is strongly motivated by natural phenomena where particles occupy more than one adsorption site. Different phase diagrams (separating the percolating and non-percolating regions) have been calculated by determining the evolution of the percolation thresholds as a function of the parameters of the problem. The main features of such phase diagrams allow to determine the expected behavior in the limit k ! 1. In order to test the universality of the problem, the phase transition involved has been studied by using finite-size scaling theory. In all the studied cases, it was established that the phase transition belongs to the random percolation universality class, as it is expected. Acknowledgments This work was supported in part by CONICET (Argentina) and the Universidad Nacional de San Luis (Argentina) under projects PIP 6294 and 322000, respectively. The numerical work were done using the BACO parallel cluster (composed by 60 PCs each with a 3.0 MHz Pentium-4 processors) located at Universidad Nacional de San Luis, San Luis, Argentina. References [1] J.M. Hammersley, Proc. Cambridge Philos. Soc. 53 (1957) 642. [2] D. Stauffer, Introduction to Percolation Theory, Taylor & Francis, London, 1985. [3] G. Deutscher, R. Zallen, J. Adler (Eds.), Percolation Structures and Processes, Annals of the Israel Physical Society, vol. 5, Ayalon Offset Ltd., 1983. [4] M. Sahimi, Application of the Percolation Theory, Taylor & Francis, London, 1992. [5] R. Zallen, The Physics of Amorphous Solids, Wiley, NY, 1983. [6] S. Kirkpatrick, Rev. Mod. Phys. 45 (1973) 574. [7] J.W. Essam, Rep. Prog. Phys. 43 (1980) 843. [8] J.-P. Hovi, A. Aharony, Phys. Rev. B 53 (1996) 235. [9] K. Binder, Rep. Prog. Phys. 60 (1997) 488. [10] Y.Y. Tarasevich, S.C. van der Marck, Int. J. Mod. Phys. C 10 (1999) 1193. [11] A. Bunde, S. Havlin, Fractal and Disordered Systems, second ed., Springer, Berlin, 1996. [12] A.R. Kerstein, J. Phys. A 16 (1983) 3071. [13] W.T. Elam, A.R. Kerstein, J.J. Rehr, Phys. Rev. Lett. 52 (1982) 1516. [14] M.D. Rintoul, Phys. Rev. B 62 (2000) 68. [15] S.C. van der Mark, Phys. Rev. Lett. 77 (1996) 1785.
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M. Dolz et al. / Physica A 374 (2007) 239–250 J. Wang, I.L. McLaughlin, M. Silbert, J. Phys. Condens. Matter 3 (1991) 5603. P. Danwanichakul, E.D. Glandt, J. Colloid Interface Sci. 283 (2005) 41. J.W. Evans, D.E. Sanders, Phys. Rev. B 39 (1989) 1587. A. Bunde, H. Harder, W. Dieterich, Solid State Ionics 18 (1986) 156. H. Harder, A. Bunde, W. Dieterich, J. Chem. Phys. 85 (1986) 4123. H. Holloway, Phys. Rev. B 37 (1988) 874. M. Nakamura, Phys. Rev. A 36 (1987) 2384. M. Henkel, F. Seno, Phys. Rev. E 53 (1996) 3662. E.L. Hinrichsen, J. Feder, T. Jossang, J. Stat. Phys. 44 (1986) 793. V. Cornette, A.J. Ramirez-Pastor, F. Nieto, Physica A 327 (2003) 71. V. Cornette, A.J. Ramirez-Pastor, F. Nieto, Eur. Phys. J. B 36 (2003) 391. M. Dolz, F. Nieto, A.J. Ramirez-Pastor, Eur. Phys. J. B 43 (2005) 363. B.E. Hayden, D.F. Klemperer, Surf. Sci. 80 (1979) 401. J.W. Evans, R.S. Nord, J. Stat. Phys. 38 (1985) 681. G.C. Barker, M.J. Grimson, Mol. Phys. 63 (1988) 145. N.M. Svrakic, M. Henkel, J. Phys. I 1 (1991) 791. J.W. Evans, Rev. Mod. Phys. 65 (1993) 1281. M. Aizenman, Nucl. Phys. B 485 (1997) 551. J. Cardy, J. Phys A 31 (1998) L105. L.N. Shchur, S.S. Kosyakov, Int. J. Mod. Phys. C 8 (1997) 473. L.N. Shchur, Incipient Spanning Clusters in Square and Cubic Percolation, in: D.P. Landau, S.P. Lewis, H.B. Schuettler (Eds.), Springer Proceedings in Physics, vol. 85, Springer, Heidelberg, Berlin, 2000. F. Yonezawa, S. Sakamoto, M. Hori, Phys. Rev. B 40 (1989) 636. F. Yonezawa, S. Sakamoto, M. Hori, Phys. Rev. B 40 (1989) 650. J. Hoshen, R. Kopelman, Phys. Rev. B 14 (1976) 3428. V. Privman, P.C. Hohenberg, A. Aharony, Universal Critical-Point Amplitude Relations, in: C. Domb, J.L. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, Academic, NY, 1991, vol. 14, pp. 1–134, 364–367, (Chapter 1.) D.W. Heermann, D. Stauffer, Z. Phys. B 40 (1980) 133. S. Fortunato, Phys. Rev. B 67 (2003) 014102; S. Fortunato, Phys. Rev. B 66 (2002) 054107. Y. Leroyer, E. Pommiers, Phys. Rev. B 50 (1994) 2795. B. Bonnier, M. Honterbeyrie, Y. Leroyer, C. Meyers, E. Pommiers, Phys. Rev. B 49 (1994) 305.
Physica A 374 (2007) 239–250 www.elsevier.com/locate/physa
Percolation processes in monomer-polyatomic mixtures M. Dolza, F. Nietob, A.J. Ramirez-Pastorb, a
Centro Ato´mico Bariloche, CONICET, Av. Bustillo 9500, 8400, S. C. de Bariloche, Argentina Departamento de Fı´sica, Universidad Nacional de San Luis, CONICET, Chacabuco 917, D5700BWS San Luis, Argentina
b
Received 13 September 2005; received in revised form 22 May 2006 Available online 21 July 2006
Abstract In this paper, the percolation of mixtures of monomers and polyatomic species (k-mers) on a square lattice is studied. By means of a finite-size scaling analysis, the critical exponents and the scaling collapsing of the fraction of percolating lattices are found. A phase diagram separating a percolating from a non-percolating region is determined. The main features of the phase diagram are discussed in order to predict its evolution for larger k-mer sizes. r 2006 Elsevier B.V. All rights reserved. Keywords: Percolation; Monte Carlo simulations; Finite-size scaling theory; Phase transitions
1. Introduction The study of clustering and percolation behavior in different physical systems is related to a wide range of phenomena of both theoretical and technological importance [1–10]. Many of these real world systems are best modeled by percolation of randomly distributed objects of a given shape and size as opposite to standard site or bond percolation in which single sites or bonds in a discrete lattice are randomly occupied. While certain quantities, i.e. critical exponents, are independent of the details of the percolation model other ones such as the percolation threshold are not [11–17]. However, most of the studies are devoted to the percolation of molecules with single occupancy. Thus, whether some sort of correlation exists, like particles occupying several ðkÞ contiguous lattice sites (k-mers), the statistical problem (multisite statistics) becomes exceedingly difficult. In this line of thinking, several authors have produced seminal contributions for analyzing the percolation of polyatomic species on different lattices [18–24]. More recently [25–27], the percolation of (a) linear segments of size k and (b) k-mers of different structures and forms deposited on a square lattice have been studied. In addition, a generalization of the pure site and pure bond percolation problems in which pairs of nearest neighbor sites (site dimers) and linear pairs of nearest neighbor bonds (bond dimers) are independently occupied at random on a square lattice was discussed in order to determine the phase diagram of the system. However, the problem is far to be exhausted. In this context, the present paper deals with the percolation of Corresponding author. Tel.: +54 2652 436151; fax: +54 2652 430224.
E-mail addresses: [email protected] (M. Dolz), [email protected] (F. Nieto), [email protected] (A.J. Ramirez-Pastor). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.06.017
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mixtures of two kind of species, namely, monomers and k-mers ðk41Þ on a regular two-dimensional lattice. Physically, this could correspond to irreversible co-adsorption of several different molecular species or, alternatively, adsorption of a single type of molecule with different binding configurations. The latter is proposed for CO adsorption on several metal surfaces. This process involves competition between two-point binding b CO at pairs of sites and one-point binding of a CO at single sites [28]. From a theoretical point of view, the competitive irreversible adsorption (or random sequential adsorption, RSA) of mixtures of monomers and k-mers has been widely studied in the literature [23,29–31]. In fact, Evans and Nord used highorder hierarchical truncation to analyze competitive filling of various small animals on a square lattice [18]. Barker and Grimson simulated RSA of mixtures of 4-mers of different shapes [30]. On the other hand, kinetics of RSA on a square lattice for mixtures of line segments of two different lengths has been studied by Svrakic and Henkel [23,31]. Other examples are compiled in Ref. [32]. A study as presented here is a natural extension of those papers, where the distributions of empty and occupied sites generated by RSA are analyzed in the framework of the percolation theory. Based in a Monte Carlo (MC) analysis, the main aim of the paper is (a) to include dynamic in the model by establishing a parameter which take into account the probability of deposition for each kind of percolating species; (b) to consider the evolution of the percolation threshold under the variation of both the relative concentration of percolation species and the size of them and (c) to determine the phase diagram as a function of the parameters of the problem in order to explain the percolative properties of the system. The paper is organized as follows. The basic definitions related to the model and the numerical techniques used in the MC analysis are presented in Section 2. In fact, in Section 2.1 the basis of the model of deposition of monomer-polyatomic mixtures on a site square lattice is given. Section 2.2 is devoted to describe the numerical method used throughout the paper, including finite-size scaling theory. The characteristic phase diagrams, along with the collapsing curves predicted by the finite-size scaling theory are shown in Section 3. Finally, conclusions are drawn in Section 4. 2. Basic definitions and details of the computer simulation 2.1. The model Let us consider a square lattice of linear size L with periodic boundary conditions. Monomers and k-mers are deposited onto such lattice until partial concentrations ð1 cÞp and cp of such particles are reached, respectively. Accordingly, (i) p represents the fraction of occupied sites, and (ii) c (ranging between 0 . . . 1) is defined as the relative concentration of k-mers that will be present in the mixture. In other words, c is the number of sites occupied by k-mers over the total number of occupied sites. The fact that one component of the mixture occupies more than one site requires the definition of a dynamic for the irreversible deposition.1 For this purpose, we include an extra parameter ðtÞ which is defined in the framework of the following scheme. First, a random number, x is selected. If x is larger than a given probability t a monomer is candidate to be deposited. Then, if both (a) the concentration of monomers is lower than ð1 cÞ and (b) a site randomly selected is vacant; the site becomes occupied. Otherwise the attempt is rejected. For values of x lower than t a k-mer is considered to be deposited whether the relative concentration of polyatomics is lower than c. In this case, a k-uple of nearest neighbor sites is randomly selected; if it is vacant, the k-mer is then dropped on those sites. Otherwise, the attempt is rejected. Thus, if t ¼ 0 the ð1 cÞpL2 monomers will be deposited first and then the L2 cp=k k-mers are dropped onto the lattice while in the opposite limit whether t ¼ 1 will be the cpL2 =k k-mers first deposited and next the corresponding ð1 cÞpL2 12 monomers. In Fig. 1 a lattice of L ¼ 6, p ¼ 19 36, c ¼ 19 and t ¼ 1 is presented for illustrative purposes where monomers (hexagons in the figure) and trimers (circles) are deposited. The central idea of the percolation theory is based in finding the minimum concentration p for which at least a cluster (a group of occupied sites in such a way that each site has at least one occupied nearest neighbor site) extends from one side to the opposite one of the system. This particular value of the concentration rate is 1
As an example, note that whether cp ½ð1 cÞp k-mers [monomers] are deposited first, the saturation limit p ¼ 1 can [cannot] be reached.
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k=1
k=3
Fig. 1. An illustrative example of the model for a square lattice of sites with size L ¼ 6. Monomers (hexagons) and trimers (circles) are 12 dropped onto the lattice with the following values of the parameters: p ¼ 19 36, c ¼ 19 and t ¼ 1.
named critical concentration or percolation threshold and determines a phase transition in the system. In the random percolation model, a single site is occupied with probability p. For the precise value of pc , the percolation threshold of sites, at least one spanning cluster connects the borders of the system [indeed, there exist a finite probability of finding n ð41Þ spanning clusters [33–36]]. In that case, a second-order phase transition appears at pc which is characterized by well defined critical exponents. As it was already mentioned, the main goal of this paper is (a) to determine how the mixture of different kinds of k-mers influences in the numerical value of the percolation threshold of the system and (b) to verify the universality class of the phase transition involved in the problem. For these purposes, long scale numerical simulations (independent of the size of the k-mer) are required in order to predict the behavior of the system in the thermodynamic limit. The ratio L=k is kept constant for avoiding spurious effects due to the k-mer size in comparison with the lattice size. A study of the finite-size effects allows us to make a reliable extrapolation to the k ! 1 limit when the limit L ! 1 is taken before. Details of this study will be given in the next subsection. 2.2. Monte Carlo simulations It is well known that it is a quite difficult matter to analytically determine the value of the percolation threshold for a given lattice [2,4,5,7,8]. For some special types of lattices, geometrical considerations enable to derive their percolation thresholds exactly. For systems which do not present such a topological advantage, percolation thresholds have to be estimated numerically by means of computer simulations. Percolation of polyatomic species adds a new ingredient to the problem: the influence of local correlation. As the scaling theory predicts [9], the larger the system size to study, the more accurate the values of the threshold obtained therefrom. Thus, the finite-size scaling theory gives us the basis to achieve the percolation threshold and the critical exponents of a system with a reasonable accuracy. For this purpose, the probability R ¼ RX L ðpÞ that a lattice composed of L L sites percolates at concentration p can be defined [2]. Here, as in Ref. [37,38], the following definitions can be given according to the meaning of X : (a) RIL ðpÞ ¼ the probability that we find a cluster which percolates both in a rightward and in a downward direction; (b) RU L ðpÞ ¼ the R 1 probability of finding either a rightward or a downward percolating cluster and (c) RA L ðpÞ 2 ½RL ðpÞ þ D I U 1 RL ðpÞ 2 ½RL ðpÞ þ RL ðpÞ. In order to determine the percolation threshold, it is necessary to evaluate the effective threshold pc ðLÞ for a lattice of finite size L. In the MC simulations, RX L ðpÞ is calculated for each discrete value of p according to the considered regular finite lattice. For determining the desired quantities two steps are required: (a) the construction of m samples for a given coverage (according to the scheme presented in Section 2.1) and (b) the
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cluster analysis by using the Hoshen and Kopelman algorithm [39]. In the last step, it is verified whether exists a percolating island. This spanning cluster could be determined by using the criteria I, U or A. n runs of such two steps are carried out for obtaining the number mX of them for which a percolating cluster of the desired X criterion X is found. Then, RX L ðpÞ ¼ m =n is defined and the procedure is repeated for different values of the parameters of the problem. A In Fig. 2a (b), the probabilities RIL ðpÞ (squares), RU L ðpÞ (circles) and RL ðpÞ (triangles) are presented for the problem of site percolation of monomers and dimers (trimers) at two different concentration as indicated and probability t ¼ 0. From a first inspection of the figure (and from data do not shown here for a sake of clarity) several conclusions are in order. Curves cross each other in a unique universal point, RX , which depends on the criterion X used. The abscissa of this point is pc ð1Þ. Those points do not modify their numerical value for the different both mixtures of polyatomic species used and values of the parameters c and t. R is also called in the literature as percolation cumulant, whose properties are identical to those of the Binder cumulant in standard thermal transitions [9,40]. Here, we have discussed the behavior of this cumulant obtained via three different criteria. One of the most important characteristics of RX shows that its value at the critical threshold is a universal quantity, i.e. is the same for models in the same universality class [41,42]. The observed findings
1.0
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A
0.57 I 0.56 0.00
(c)
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0.04
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Fig. 2. (a) Fraction of percolating lattices as a function p. Different criteria are used to identify the spanning cluster, namely, RU L ðpÞ of the (circles); RIL ðpÞ (squares) and RA L ðpÞ (triangles). This example corresponds to the mixture monomer–dimer and the particular values parameters c ¼ 0:95 (full symbols) and c ¼ 0:20 (empty symbols) and t ¼ 0 for both cases. Horizontal dashed lines show the RX universal points. Vertical dashed lines denote the percolation threshold, pc in the thermodynamic limit L ! 1. (b) As Fig. 2(a) for the mixture monomer–trimer and the parameters of the problem as indicated. (c) Extrapolation of pc ðLÞ of Fig. 2a towards the thermodynamic limit according to the theoretical prediction given by Eq. (1). Squares, triangles and circles denote the values of pc ðLÞ obtained by using the criteria I, A and U, respectively. The error bars are smaller that the symbol size.
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are then a clear indication (as it is expected) that the problem belongs to the same universality class regardless both the mixture considered and the values of the parameter of the model. X For each curve RX L ðpÞ, pc ðLÞ can be obtained from the value in which the slope is maximum. The X extrapolation of pc ðLÞ toward the limit L ! 1 is predicted to follow: X 1=n , pX c ðLÞ ¼ pc ð1Þ þ A L
(1)
X
where A is a non-universal constant and n is the critical exponent associated with the correlation length which is analytically shown to be equal to n ¼ 43 in the case of random percolation. Fig. 2c shows the extrapolation towards the thermodynamic limit of pX c ðLÞ according to Eq. (1) for the parameters used in Fig. 2a corresponding to c ¼ 0:95. This figure lends support to the assertion given by Eq. (1): (a) all the curves (different criteria) are well correlated by a linear function, (b) they have a quite similar value for the ordinate in the limit L ! 1 and (c) the fitting determines a different value of the constant A depending of the type of criterion used. It is also important to note that pA c ðLÞ gives a perfect horizontal line which is a great advantage of the method because it does not require precise values of critical exponents in the process of estimating U A percolation thresholds. The maximum of the differences between jpIc ð1Þ pA c ð1Þj and jpc ð1Þ pc ð1Þj gives the error bar for each determination of pc . For technical details in the application of the method, (in particular, for determining the critical exponents) we refer the interested reader to Refs. [25–27,37,38]. 3. Results and discussions 3.1. Phase diagrams We focus now on the influence of the filling dynamics on the percolation threshold upon varying the probability t from 0 to 1 while c remains constant. These curves are shown in Fig. 3 for a monomer-dimer mixture and different values of c, as indicated. Several considerations are in order about this behavior. In particular, two different regimes can be distinguished. The limit t ¼ 0 reflects the situation where all the particles dropped first are monomers. Then, the k-mers are deposited in the vacant sites. Upon increasing smoothly t a increasing fraction of k-mers are deposited when not all the monomers are dropped. The local correlation introduced by the k-mers produce more compact structures which in turn slightly reduce the percolation threshold. In the opposite limit, for t ¼ 1 the interstitial holes needed to be filled for building the spanning cluster must be occupied by monomers. This situation is favorable as compared with the case t ¼ 0 where k-mers must fill the vacant sites and a larger value of surface coverage is needed for connecting the extremes of the lattice being always pc ðt ¼ 0Þ4pc ðt ¼ 1Þ. The above physical considerations constitute two
c=0.15
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0.565 0.0
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Fig. 3. Percolation thresholds as a function of t for several values of the constant c, as indicated. The curves shown such behavior for a monomer–dimer mixture.
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different regimes for the limit cases t ¼ 0 and t ¼ 1. The curves show a inflexion point between these two different regimes approximately when t c. The numerical value of pc ðtÞ remains almost constant far away of t c, i.e. ( pc ð0Þ if t5c pc ðtÞ (2) pc ð1Þ if tbc: Data from Fig. 3 permit to build Fig. 4 where the percolation thresholds are now plotted as a function of c being the curves parametrized by t. Three regions can be distinguished. The first one corresponds to the nonpercolating (NP) area characterized by the relationship pðc; tÞopc ðc; 1Þ. In a second region where pðc; tÞ4pc ðc; 0Þ, the system always percolates and is labeled as P1 in the figure. Finally, the dashed area P2 corresponds to surface concentrations which percolate depending of the value of t, for each c, according to Fig. 3. pc ðc; t ¼ 1Þ is a monotonically decreasing curve. Upon increasing the concentration of k-mers in the lattice the percolation threshold decreases as a consequence of the local correlation in agreement with Refs. [25,26,43,44]. However, the curve pc ðc; t ¼ 0Þ increases as c is raised, then goes through a definite maximum and finally decreases for larger values of c. This behavior can be understood as follows. For c ¼ 0 (all the particles on the lattice are monomers) the percolation threshold is the well-known value pc ¼ 0:5927 [2]. In this conditions, the minimum increasing of c corresponds to an exchange of one monomer by one dimer. This procedure clearly deals to a new critical concentration larger than the percolation threshold for the case of monomers ðc ¼ 0Þ. Obviously, this effect is enhanced for larger values of k. However, as c increases, the local correlation present due to the adsorption of k-mers limits the above described tendency. Then, the curve reaches a defined maximum and monotonically diminishes down to pc ð1; 0Þ. This extreme value pc ðc ¼ 1; t ¼ 0Þ ¼ pc ðc ¼ 1; t ¼ 1Þ is the percolation threshold of dimers on a square lattice [25,26]. Figs. 3 and 4 represent the case of monomer-dimer mixtures while Fig. 5 shows the evolution of those curves for different monomerk-mer mixtures, as indicated. For all the curves considered, the initial point is the same pc ðc ¼ 0; tÞ ¼ 0:5927. In the other extreme, c ¼ 1 the percolation threshold is only a function of k which is given by pc ðc ¼ 1; t ¼ 0Þ ¼ pc ðc ¼ 1; t ¼ 1Þ ¼ pc ðkÞ. The numerical values of such function have been obtained in Refs. [25,26] where it was shown that pc ðk ! 1Þ 0:461. In order to predict the evolution of these curves in the thermodynamic limit k ! 1, we consider the relevant features of them. The area p, defined as: p ¼ \P2 ,
(3)
t=0 t=1
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pc (c)
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NP 0.57
0.56 0.0
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Fig. 4. Percolation thresholds are plotted as a function of c while t remains constant. Three regions can be distinguished: (a) the nonpercolating (NP) area characterized by the relationship pðc; tÞopc ðc; 1Þ; (b) P1 , where the system always percolates, pðc; tÞ4pc ðc; 0Þ and (c) the dashed area P2 , where the system percolates depending of the value of t for each c. The curves show the behavior for a monomer–dimer mixture.
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Fig. 5. Idem Fig. 4 for different mixtures, as indicated. The quantity p, defined through Eq. (3), is represented by the dashed area.
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Fig. 6. (a) Evolution of the area p as function of the k-mer size. (b) Difference Dpc pc ðc; t ¼ 0Þ pc ðc; t ¼ 1Þ versus c for several monomer-k-mers mixtures.
is represented as a dashed region in Fig. 5 and its evolution is shown in Fig. 6a as a function of the k-mers size. The dots fit very well with a decreasing exponential function whose limit value is pðk ! 1Þ 0:0017. On the other hand, the difference Dpc pc ðc; t ¼ 0Þ pc ðc; t ¼ 1Þ is plotted as a function of c for several monomer-kmers mixtures, as indicated in Fig. 6b. It is observed that Dpc tends to a limiting curve as the size of the k-mers
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increases. By considering both features, the curve with spheres in Fig. 7 is postulated as the response of the system in the thermodynamic limit k ! 1. Finally, a limiting curve separating the percolating and the NP region independently of both: (a) the filling mechanism (in other words, regardless the value of t) and; (b) the k-mer size can be defined. Thus, for each value of c, the largest value of the percolation threshold, pc ðc; t ¼ 0; kÞ, is collected in a single curve. The result of the procedure is shown in Fig. 8. 3.2. Finite-size scaling analysis In this subsection, the influence of the parameters c and k on the collapsing curves is studied in the light of the finite-size scaling theory. The scaling law hypothesis predicts the collapsing of the curves RX L ðpÞ when they
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C Fig. 7. Idem Fig. 4 for different mixtures. The prediction of the limiting curve for k ! 1 is included. p1 represent the tendency of p for k ! 1 according to Fig. 6b.
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Fig. 8. The limiting curves separating the percolating and the non-percolating regions independently of both (a) the filling mechanism and (b) the k-mer size. Indicated portions of the curve correspond to different monomer-k-mer mixtures.
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are plotted as a function of a reduced variable u ðp pc ÞL1=n : 1=n X RX , L ðpÞ ¼ R ½ðp pc ÞL
(4)
being RX ðuÞ the scaling function. Thus, RX is an universal function with respect to the variable u. In Fig. 9, as an illustration, we plot RA L as a function of u for (a) monomer–dimer; (b) monomer–trimer and (c) monomer–tetramer mixtures and several values of concentration as indicated. The collapse gives an additional indication for the numerical value of the critical exponent n. As it is clearly seen from this analysis, the problem belongs to the same universality class of random percolation regardless of the value of c and k considered. In all cases, the curves nicely collapse into a unique universal curve according to the theoretical prediction. However, the scaling function seems to be different for each mixture considered. In Fig. 10, a representative curve of each mixture has been plotted as a function of u. This fact determines that the scaling function RX is not an universal function with respect to the variable k (each value of k is represented by using a different symbol, as indicated). 1.0 (a) 0.10
RAL (P)
0.20 0.30
0.5
0.40 0.50 0.60 0.80 0.90
0.0 (b)
0.10 0.20
RAL (P)
0.30 0.40
0.5
0.50 0.60 0.70 0.80 0.90
0.0 (c)
RAL (P)
0.10 0.20 0.30
0.5
0.40 0.50 0.60 0.70
0.0 -1.0
-0.5
0.0 (p-pc)L1/v
0.5
1.0
1=n Fig. 9. RA for (a) monomer–dimer; (b) monomer–trimer and (c) monomer–tetramer mixtures and several L as a function of u ðp pc ÞL values of concentration c.
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248 1.0
1-2 1-3 1-4 1-5 1-6 1-7
RAL (P)
0.8
0.6
0.4
0.2
0.0 -2
-1
0 (p-pc)L1/v k
1
2
Fig. 10. Representative curves of several mixtures plotted as a function of u.
In order to determine the dependence of RX with k, the main features of the collapsing data have been considered in the range of k between 2 and 7. The curves become steeper upon increasing the value of k. Thus, we can observe that: (a) The derivatives become more pronounced as k increases. It is possible to establish a power-law to describe this behavior, ! qRX ¼ Bkr . (5) qu max
In a log–log scale the points are very well correlated by a linear function, like in Eq. (5), being the numerical values of the fitting parameters very similar for the three criteria used here. (b) The derivatives are narrowed upon increasing k. This behavior can also be described by a power-law according to: DX ¼ Ckl , X
(6) X
where D is the standard deviation of ðqR =quÞ for each curve. Thus, the standard deviation of each derivative versus k, when is plotted in a log–log scale, is very well correlated by a linear function, being the fitting parameter l ¼ 0:95 0:02 for A, U and I criteria (not shown here). According to the equations above, a metric factor might be included in the scaling function, Eq. (4), in order to collapse all the curves in Fig. 10 onto a single one. Following Ref. [40], in Fig. 11 we plot the probability RX L as a function of the argument z ¼ ðp pc ÞL1=n kl . The curves nicely collapse in the close vicinity of z ¼ 0 (close to the critical point) and a tiny deviation is observed as jzj increases. The scaling analysis given above should be rigorously valid only for sufficiently large L and from p in the asymptotic critical regime. However, it can be applied to the entire range of L and p if the data falls in the ‘‘domain of attraction’’ of a simple fixed point characterizing only one universality class of critical phenomena [9]. It is remarkable that more than 3 103 points are included in the collapsing curve. The metric factor introduced here, kl , gives an additional indication for the numerical value of the exponent l obtained in Eq. (6). 4. Conclusions In this paper, we have studied the percolation process for monomer-k-mer mixtures with increasing values of k. The percolating species, monomers and k-mers, are deposited at different relative concentrations, (1 c)
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1.0 1-2 1-3
0.8
RAL (P)
1-4 1-5
0.6
1-6 1-7
0.4
0.2
0.0 -5
-4
-3
-2
-1
0
1
2
3
4
5
(p-pc)L1/v k 1=n l Fig. 11. RA k , where l is obtained from Eq. (6). L as a function of the argument z ¼ ðp pc ÞL
and c, respectively. In addition, we have taken into account different probabilities of deposition for each species. This study does not attempt to reproduce any particular experimental situation but it is strongly motivated by natural phenomena where particles occupy more than one adsorption site. Different phase diagrams (separating the percolating and non-percolating regions) have been calculated by determining the evolution of the percolation thresholds as a function of the parameters of the problem. The main features of such phase diagrams allow to determine the expected behavior in the limit k ! 1. In order to test the universality of the problem, the phase transition involved has been studied by using finite-size scaling theory. In all the studied cases, it was established that the phase transition belongs to the random percolation universality class, as it is expected. Acknowledgments This work was supported in part by CONICET (Argentina) and the Universidad Nacional de San Luis (Argentina) under projects PIP 6294 and 322000, respectively. The numerical work were done using the BACO parallel cluster (composed by 60 PCs each with a 3.0 MHz Pentium-4 processors) located at Universidad Nacional de San Luis, San Luis, Argentina. References [1] J.M. Hammersley, Proc. Cambridge Philos. Soc. 53 (1957) 642. [2] D. Stauffer, Introduction to Percolation Theory, Taylor & Francis, London, 1985. [3] G. Deutscher, R. Zallen, J. Adler (Eds.), Percolation Structures and Processes, Annals of the Israel Physical Society, vol. 5, Ayalon Offset Ltd., 1983. [4] M. Sahimi, Application of the Percolation Theory, Taylor & Francis, London, 1992. [5] R. Zallen, The Physics of Amorphous Solids, Wiley, NY, 1983. [6] S. Kirkpatrick, Rev. Mod. Phys. 45 (1973) 574. [7] J.W. Essam, Rep. Prog. Phys. 43 (1980) 843. [8] J.-P. Hovi, A. Aharony, Phys. Rev. B 53 (1996) 235. [9] K. Binder, Rep. Prog. Phys. 60 (1997) 488. [10] Y.Y. Tarasevich, S.C. van der Marck, Int. J. Mod. Phys. C 10 (1999) 1193. [11] A. Bunde, S. Havlin, Fractal and Disordered Systems, second ed., Springer, Berlin, 1996. [12] A.R. Kerstein, J. Phys. A 16 (1983) 3071. [13] W.T. Elam, A.R. Kerstein, J.J. Rehr, Phys. Rev. Lett. 52 (1982) 1516. [14] M.D. Rintoul, Phys. Rev. B 62 (2000) 68. [15] S.C. van der Mark, Phys. Rev. Lett. 77 (1996) 1785.
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