Percolation of the aligned dimers on a square lattice
V.A. Cherkasova, Yu.Yu. Tarasevich, N.I. Lebovka, N.V. Vygornitskii
aa r X i v : . [ c ond - m a t . d i s - nn ] D ec EPJ manuscript No. (will be inserted by the editor)
Percolation of the aligned dimers on a square lattice
V.A. Cherkasova , Yu.Yu. Tarasevich , N.I. Lebovka , and N.V. Vygornitskii Astrakhan State University, 20a Tatishchev Str, Astrakhan, 414056, Russia Institute of Biocolloidal Chemistry named after F.D. Ovcharenko, NAS of Ukraine, 42, blvr. Vernadskogo, Kyiv, 03142,Ukraine Received: date / Revised version: date
Abstract.
Percolation and jamming phenomena are investigated for anisotropic sequential deposition ofdimers (particles occupying two adjacent adsorption sites) on a square lattice. The influence of dimeralignment on the electrical conductivity was examined. The percolation threshold for deposition of dimerswas lower than for deposition of monomers. Nevertheless, the problem belongs to the universality class ofrandom percolation. The lowest percolation threshold ( p c = 0 . p c = 0 . PACS.
Physical properties of (partially) disordered systems aredescribed in percolation approach [1,2,3,4,5,6,7]. Effect ofconcentration on physical properties is well understood.Nevertheless, alignment effect plays essential role for highaspect ratio objects, such as nanotubes and nanowires [8].Percolating properties and behaviors of systems, com-posed of anisotropic nanoparticles, are extensively investi-gated during last years [9,10,11,12,13,14]. The problem ofanisotropic percolation has been a subject of many inves-tigations. It is of interest and of value to inquire how anorientation of nanoparticles influences the main physicalproperties of the systems. Effect of the nanotube align-ment on conductivity is of specific interest. The anisotropycan be induced by various factors. For instance, alignmentof the nanotubes may be induced by flow [8] or electricfield [15,16].The electrical properties of a ceramic composition wereconsidered in [12]. A profound influence of the conduct-ing phase structure on the electrical conductivity of thematerial was shown. The simulations indicated consider-able dependence of the percolation cluster properties onanisotropy of its components.The irreversible adsorption (deposition) of particles onsolid surfaces is a subject of considerable practical im-portance. A well known example of an irreversible mono-layer deposition process is the random sequential adsorp-tion (RSA). RSA has attracted significant interest due toits importance in many physical, chemical, and biologi-cal processes. RSA is a natural model for irreversible and sequential deposition of macromolecules at solid–liquid in-terfaces. Some examples of the wide range of applicabilityof this model include adhesion of colloidal particles, as wellas adsorption of proteins to solid surfaces, with relaxationtimes much longer than the deposit formation time. Thisprocess is well described in the literature and has beeninvestigated extensively in the last decades. The topic hasbeen well covered in [17,18].Investigation of the irreversible adsorption of polyatomicspecies ( k -mers) has received considerable attention in thelast years. The results of the study of random sequentialadsorption and percolation of polyatomic species on differ-ent substrates were presented in [13,19,20,21,22,23,24,25,26,27,28].Numerical studies of random sequential adsorption (theRSA model) of rectangular particles on a flat substrateare performed and dependencies of the saturation concen-tration and the percolation threshold on the model pa-rameters are determined in [14].The main goal of the present study is to investigatethe influence of alignment on the percolation and jam-ming thresholds, as well as conductivity. In this paper, weprovide accurate numerical data for the percolation andjamming thresholds of the (partially) ordered (aligned)dimers on a square lattice. We perform calculation of elec-trical conductivity as a function of dimers alignment. Theproposed model can be useful for description of the per-colation behaviour of anisotropic conductor networks.The paper is organized as follows. In Section 2 the ba-sis of the model of deposition of dimers on a square latticeis presented. of The results, obtained using the finite sizescaling theory, are also analysed and discussed in this Sec- V.A. Cherkasova et al.: Percolation of the aligned dimers on a square lattice tion. The results, related to conductivity of the systemsconcerned, are presented in Section 3 pres. Finally, we dis-cuss dependence of the percolation threshold on the modelparameters of interest in Section 4.
Different kinds of boundary conditions were used in perco-lation simulations: open, periodic or toroidal, and cylin-drical, i.e. open along one direction and periodic alonganother one. One uses usually the term crossing only tosystems with open boundaries, and the terms spanningand wrapping for cylindrical boundary conditions [29]. Inour study, the periodic boundary conditions are appliedby gluing along the vertical and horizontal borders. Wedefine a wrapping cluster as a cluster that winds aroundthe system along the given direction, i.e. it provides a pathwith the length of 2 π [29].Mersenne Twister random number generator [30] wasutilized for filling in the lattice with dimer at given con-centration and orientation. It has a period of 2 − s : s = N | − N − N | + N − , where N | and N − are the numbers of dimers, oriented invertical and horizontal direction, respectively [31].Let us consider a periodic square lattice of linear size L , on which dimers are deposited at random, but withgiven orientation. Two of nearest neighbour sites alonggiven direction are randomly selected; if both sites arevacant, the dimer adsorbs on such sites. Otherwise, theattempt is rejected. In any case, the procedure is iterateduntil N = N | + N − dimers are adsorbed and the desiredconcentration is reached.We perform our calculations of percolation thresholdsusing the Hoshen–Kopelman algorithm [32].Fig. 1 demonstrates percolation in an anisotropic sys-tem of dimers (the dimers are oriented strictly along thevertical direction). Fig. 1.
Color online. Percolation on a square lattice of linearsize L = 64 at s = 1 (wrapping cluster is indicated in red) The final state, generated by irreversible adsorption, isa disordered state (known as jamming state), in which nomore objects can be deposited due to the absence of anyfree space of appropriate size and shape [17]. If differentorientations of the deposited objects are not equally prob-able, the definition of the jamming state is to be refined.Let us assume N | > N − . We define jamming for the fixedparameter s as a situation when there is no possibility ofdepositing any additional vertically oriented object. Nev-ertheless, there may be places for accepting horizontallyoriented objects.Fig. 2 shows jamming for different s at a lattice oflinear size L = 64. Fig. 2.
Color online. Jamming at a lattice of linear size 64 at s = − s = 0 (right) (horizontally oriented dimersare shown in red, whereas the vertically oriented objects areshown in blue ) The percolation probability P vs. the site occupationprobability p was obtained for the lattices of linear sizes L = 64 , , , P as a function of occupa-tion p for a particular value of s is shown in Fig. 3. Thepercolation threshold p c for a given lattice size L can beestimated from the condition P ( p ) = 0 . P ( p ) p Fig. 3.
Percolation probability P vs. occupation probability p for a lattice of linear size L = 1024 at s = 0 . p c = 0 . ± . The finite-size scaling analysis was carried out for get-ting the percolation threshold at L → ∞ . The percolationthreshold p c ( L ) was calculated for five different values of .A. Cherkasova et al.: Percolation of the aligned dimers on a square lattice 3 the linear lattice size. The percolation threshold p c ( ∞ ) ofan infinite lattice can be found by fitting the results forlattices of different sizes to the scaling relation (Fig. 4): | p c ( L ) − p c ( ∞ ) | ∝ L − /ν , (1)where the critical exponent ν has the value 4 / p c = 0 . s = 1), and p c = 0 . s = 0). The last result agrees with theknown results [22,28] (see Table. 1). Table 1.
Comparison of published and our results for dimersoriented in two directions with equal probability p c p jam p c /p jam Our result 0 . . . L max = 2000 [22] 0 .
562 0 .
906 0 , ± . L max = 112, 5 × runs [28] 0 .
562 0 .
907 0 . L -1/ ν p c Fig. 4.
Example of percolation threshold determination in thethermodynamical limit of ( L → ∞ ) using the scaling rela-tion (1) ( s = 0, p c = 0 . ± . Moreover, several quantities of interest [6] were calcu-lated.Average cluster size: S = X i n i i / X i n i i, (2)where n i is the average number of i clusters per latticesite.The strength of the infinite network P ∞ , i.e. the proba-bility of an arbitrary site belonging to the infinite network: P ∞ ( p ) = N ∞ N , (3)where N is the total number of sites. The average cluster size S and strength of the infinitenetwork P ∞ yield the scaling laws: S ( p ) ∝ | p − p c | − γ , (4)and P ∞ ( p ) ∝ ( p − p c ) β , (5)where γ and β the universal critical exponents. In d =2 [35]: γ = 2 . , (6)and β = 0 . . (7)In our computations, the average cluster size (2) andthe strength of the infinite network (3) demonstrate typ-ical behavior near the percolation transition. The criticalexponents γ = 2 . ± .
01, at p < p c , γ = 2 . ± . p > p c , and β = 0 . ± .
002 for s = 0, extractedfrom the power laws (4) and (5), are close to the knownvalues (6) and (7).The fractal dimension of the incipient cluster [6], cal-culated as d f = d − βν , is close to the known values d f = 1 . d f = 1 . s = 0 .We found that parabolic law is a reasonable fit for p c vs. s (Fig. 5).The jamming threshold for L → ∞ was found fromscaling relation (1), where ν = 1 . ± . s = ±
1) (in fact, this case is quite equivalent to jamming inone dimension) p jam = 0 . p jam = 1 − e − ≈ . p jam ( s ). -1,2 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 1,20,550,560,570,580,590,840,860,880,900,92 s p c p j a m Fig. 5.
Percolation threshold p c vs. orientation order param-eter s , p c ( s ) = 0 . s + 0 . s , p jam ( s ) = − . s − . | s | + 0 . The highly efficient algorithm, proposed by Frank andLobb [33] was utilized for finding conductivity of a squarelattice, filled with the dimers.The Frank and Lobb algorithm utilises the repeatedapplication of a sequence of series, parallel and star-triangle(Y– △ ) transformations to the bonds of the lattice. The fi-nal result of this sequence of transformations is reductionof any finite portion of the lattice to a single bond thathas the same conductance as the entire lattice. We usedfour equivalent resistors (conductors) with σ f = 10 and σ i = 1 for occupied and empty sites [34], respectively,instead of each cell (see Fig. 6). s f s i dimer s f s i dimer Fig. 6.
Equivalent schema of a square lattice filled with dimers,where σ f and σ i are conductivity of the occupied side ( σ f =10 ) and empty one ( σ i = 1), respectively σ p p p c σ s p σ s s Fig. 7.
Conductivity σ vs. occupation probability p forisotropic ( s = 0)and strictly oriented dimers along the fixeddirection ( s = ± L = 256 Examples of conductivity σ as a function of occupationprobability p at different values of parameter s are shownin Fig. 7. Behavior of the conductivity σ ( p ) correspondswith direct estimations of the percolation threshold p c . There is a sharp transition of conductivity from σ − to σ + near the percolation threshold (see right inset in Fig. 7).Moreover, σ increases on passing from isotropic ( s = 0) tostrictly ordered dimers ( s = ± s L σ + , - σ + σ - Fig. 8.
Application of scaling to calculation of the ratio ofconductivity exponent to the exponent of the correlation lengthfor isotropic ( s = 0) and strictly oriented ( s = ±
1) dimers.Dashed line corresponds to the slope 0.973, which is typicalfor percolation in two dimensions [36]
It is known that conductivity σ behaviour near the per-colation threshold ( p − p c ≪
1) obeys the scaling relations[6]: σ − ∝ ( p c − p ) − s c , p < p c , (8) σ + ∝ ( p − p c ) t c , p > p c , (9)where t c , s c are the conductivity exponents.A detailed study of the finite size effects is presentedin order to discuss the universality class of the phase tran-sition which the system undergoes. The main aim of thepaper is to determine the dependenceFor purposes of checking the universality class and cal-culating the values of t c , s c ,the relations (8) and (9) canbe written in the following form σ − ∝ L s c /ν , p < p c , (10) σ + ∝ L − t c /ν , p > p c , (11)where L is the linear lattice size [36].Fig. 8 shows results of the finite-size analyzes of σ − and σ + . Ratios of the conductivity exponents to the cor-relation length exponent t c /ν and s c /ν are independentof s and close to 0 . ± .
05, which is a typical value forthe percolation in d = 2 [36]. Hence, our model belongs tothe class of random percolation in two dimensions. Thisstatement agrees with the conclusion drawn from analysisof the exponents ν , γ , β .Near the percolation point, behaviour of the conduc-tivity depends essentially on s . At given p , the conductiv-ity can fall down near the percolation threshold ( p → p c )if alignment of dimers ( | s | increases). Thus, alignment in-duces decrease of the conductivity at given conditions. .A. Cherkasova et al.: Percolation of the aligned dimers on a square lattice 5 -1 -0.5 0 0.5 11.522.5 s s k p -1+1 ( σ - σ i ) / p Fig. 9.
Intrinsic conductivity k vs. parameter s . In the inset:( σ/σ i − /p vs. p for s = 0 , ± L → ∞ . [36]. However, the situation changes drastically if occupa-tion probability is rather small p < .
1. At given p , in-crease of s leads to decrease of σ . The conductivity ismaximum when all the dimers are aligned along the con-ductivity direction (see left inset in Fig. 7). If p is smallenough, the conductivity behaviour can be well fitted byvirial expansion [37] σ/σ i = 1 + kp + mp + . . . , (12)where k and m are the adjustable parameters.The value of intrinsic conductivity k can be extractedfrom ( σ/σ i − /p vs. p in the limit of p → k as a function of orientation or-der parameter s is fitted well by linear function k = 1 . . s ( with correlation ρ = 0 . . ± . Thus, new percolation problem, i.e. percolation of aligneddimers on a square lattice, was proposed and studied. Thepercolation threshold for deposition of dimers was lowerthan for deposition of monomer, p c = 0 . . . . , never-theless, the problem belongs to the same universality class.The lowest percolation threshold p c = 0 .
562 corre-sponds to isotropic orientation of the dimers ( s = 0).In the case of dimers aligned strictly along one direction( s = +1 , − p c = 0 . p → p c ), the conduc-tivity essentially decreases if the absolute value of orienta-tion order parameter | s | increases. Intrinsic conductivity k increases linearly with s ( k = 1 .
88 + 0 . s ) and differsfrom the known value for a lattice filled with monomers k = 1 . ± .
01. The proposed model can be applied to the phase tran-sitions of anisotropic objects on a lattice when their con-centration and orientation are varied. In particular, themodel is useful for description of a phase transition frominsulator to (semi)conductor upon aligned deposition ofthe prolate objects on a substrate. The natural extensionof the model is substitution of dimers by k -mers and inclu-sion of the bonds between the dimers into consideration. This work has been supported by Russian Foundation forBasic Research (grant no. 09-02-90440) and Ministry ofEducation and Science of Ukraine (Project no F28.2/058).
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