Jamming and percolation of parallel squares in single-cluster growth model
This work studies the jamming and percolation of parallel squares in a single-cluster growth model. The Leath-Alexandrowicz method was used to grow a cluster from an active seed site. The sites of a square lattice were occupied by addition of the equal size k x k squares (E-problem) or a mixture of...
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| Cite this: | Jamming and percolation of parallel squares in single-cluster growth model / I.A. Kriuchevskyi, L.A. Bulavin, Yu.Yu. Tarasevich, N.I. Lebovka // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33006:1-11. — Бібліогр.: 42 назв.— англ. |
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Kriuchevskyi, I.A. Bulavin, L.A. Tarasevich, Yu.Yu. Lebovka, N.I. 2019-06-14T10:26:40Z 2019-06-14T10:26:40Z 2014 Jamming and percolation of parallel squares in single-cluster growth model / I.A. Kriuchevskyi, L.A. Bulavin, Yu.Yu. Tarasevich, N.I. Lebovka // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33006:1-11. — Бібліогр.: 42 назв.— англ. 1607-324X DOI:10.5488/CMP.17.33006 PACS: 02.70.Uu, 05.65.+b, 36.40.Mr, 61.46.Bc, 64.60.ah arXiv:1410.4292 https://nasplib.isofts.kiev.ua/handle/123456789/153448 This work studies the jamming and percolation of parallel squares in a single-cluster growth model. The Leath-Alexandrowicz method was used to grow a cluster from an active seed site. The sites of a square lattice were occupied by addition of the equal size k x k squares (E-problem) or a mixture of k x k and m x m (m ≤ k) squares (M-problem). The larger k x k squares were assumed to be active (conductive) and the smaller m x m squares were assumed to be blocked (non-conductive). For equal size k x k squares (E-problem) the value of pj = 0.638 ± 0.001 was obtained for the jamming concentration in the limit of k →∞. This value was noticeably larger than that previously reported for a random sequential adsorption model, pj = 0.564 ± 0.002. It was observed that the value of percolation threshold pc (i.e., the ratio of the area of active k x k squares and the total area of k x k squares in the percolation point) increased with an increase of k. For mixture of k x k and m x m squares (M-problem), the value of pc noticeably increased with an increase of k at a fixed value of m and approached 1 at k ≥ 10 m. This reflects that percolation of larger active squares in M-problem can be effectively suppressed in the presence of smaller blocked squares. В роботi вивчено явища джамiнгу i перколяцiї паралельних квадратiв для однокластерної моделi росту. Для росту кластеру з активного зародку використовувався метод Лiса-Александровича. Вузли квадратної ґратки займалися додаванням однакових k ×k квадратiв (E-задача) або сумiшi k ×k i m ×m (m É k) квадратiв (M-задача). Припускалося, що бiльшi k × k областi були активними (провiдними), а меншi були заблокованими (непровiдними). Для k ×k квадратiв однакового розмiру (E-задача) за умови k → ∞ було отримано таке значення концентрацiї джамiнгу p j = 0.638±0.001 . Це значення було iстотно меншим за отримане ранiше для моделi випадкової послiдовної адсорбцiї: p j = 0.564±0.002. Було показано, що величина перколяцiйного порогу pc (тобто вiдношення площi активних k ×k квадратiв до загальної площi осаджених k × k квадратiв в перколяцiйнiй точцi) зростала при збiльшеннi k. Для сумiшi k × k i m × m квадратiв (M-задача) величина pc сильно зростала при збiльшеннi k при фiксованому значеннi m та наближалась до 1 приk Ê 10m. Це пов’язано з тим, що перколяцiя бiльших активних квадратiв для M-задачi може ефективно пригнiчуватися за наявностi невеликої кiлькостi малих заблокованих квадратiв. Authors would like to acknowledge the partial financial support of project 43–02–14(U), Ukraine (N.L.) and of project RFBR 14–02–90402_Ukr, Russia (Yu.T.). Authors also thank Dr. N.S. Pivovarova for her help with the preparation of the manuscript. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Jamming and percolation of parallel squares in single-cluster growth model Джамiнг та перколяцiя паралельних квадратiв в однокластернiй моделi росту Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Jamming and percolation of parallel squares in single-cluster growth model |
| spellingShingle |
Jamming and percolation of parallel squares in single-cluster growth model Kriuchevskyi, I.A. Bulavin, L.A. Tarasevich, Yu.Yu. Lebovka, N.I. |
| title_short |
Jamming and percolation of parallel squares in single-cluster growth model |
| title_full |
Jamming and percolation of parallel squares in single-cluster growth model |
| title_fullStr |
Jamming and percolation of parallel squares in single-cluster growth model |
| title_full_unstemmed |
Jamming and percolation of parallel squares in single-cluster growth model |
| title_sort |
jamming and percolation of parallel squares in single-cluster growth model |
| author |
Kriuchevskyi, I.A. Bulavin, L.A. Tarasevich, Yu.Yu. Lebovka, N.I. |
| author_facet |
Kriuchevskyi, I.A. Bulavin, L.A. Tarasevich, Yu.Yu. Lebovka, N.I. |
| publishDate |
2014 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Джамiнг та перколяцiя паралельних квадратiв в однокластернiй моделi росту |
| description |
This work studies the jamming and percolation of parallel squares in a single-cluster growth model. The Leath-Alexandrowicz method was used to grow a cluster from an active seed site. The sites of a square lattice were occupied by addition of the equal size k x k squares (E-problem) or a mixture of k x k and m x m (m ≤ k) squares (M-problem). The larger k x k squares were assumed to be active (conductive) and the smaller m x m squares were assumed to be blocked (non-conductive). For equal size k x k squares (E-problem) the value of pj = 0.638 ± 0.001 was obtained for the jamming concentration in the limit of k →∞. This value was noticeably larger than that previously reported for a random sequential adsorption model, pj = 0.564 ± 0.002. It was observed that the value of percolation threshold pc (i.e., the ratio of the area of active k x k squares and the total area of k x k squares in the percolation point) increased with an increase of k. For mixture of k x k and m x m squares (M-problem), the value of pc noticeably increased with an increase of k at a fixed value of m and approached 1 at k ≥ 10 m. This reflects that percolation of larger active squares in M-problem can be effectively suppressed in the presence of smaller blocked squares.
В роботi вивчено явища джамiнгу i перколяцiї паралельних квадратiв для однокластерної моделi росту.
Для росту кластеру з активного зародку використовувався метод Лiса-Александровича. Вузли квадратної
ґратки займалися додаванням однакових k ×k квадратiв (E-задача) або сумiшi k ×k i m ×m (m É k) квадратiв (M-задача). Припускалося, що бiльшi k × k областi були активними (провiдними), а меншi були
заблокованими (непровiдними). Для k ×k квадратiв однакового розмiру (E-задача) за умови k → ∞ було
отримано таке значення концентрацiї джамiнгу p j = 0.638±0.001 . Це значення було iстотно меншим за
отримане ранiше для моделi випадкової послiдовної адсорбцiї: p j = 0.564±0.002. Було показано, що величина перколяцiйного порогу pc (тобто вiдношення площi активних k ×k квадратiв до загальної площi
осаджених k × k квадратiв в перколяцiйнiй точцi) зростала при збiльшеннi k. Для сумiшi k × k i m × m
квадратiв (M-задача) величина pc сильно зростала при збiльшеннi k при фiксованому значеннi m та наближалась до 1 приk Ê 10m. Це пов’язано з тим, що перколяцiя бiльших активних квадратiв для M-задачi
може ефективно пригнiчуватися за наявностi невеликої кiлькостi малих заблокованих квадратiв.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/153448 |
| citation_txt |
Jamming and percolation of parallel squares in single-cluster growth model / I.A. Kriuchevskyi, L.A. Bulavin, Yu.Yu. Tarasevich, N.I. Lebovka // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33006:1-11. — Бібліогр.: 42 назв.— англ. |
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2025-11-25T22:33:38Z |
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2025-11-25T22:33:38Z |
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| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 3, 33006: 1–11
DOI: 10.5488/CMP.17.33006
http://www.icmp.lviv.ua/journal
Jamming and percolation of parallel squares in
single-cluster growth model∗
I.A. Kriuchevskyi1, L.A. Bulavin1, Yu.Yu. Tarasevich2, N.I. Lebovka3†
1 Taras Shevchenko Kiev National University, Department of Physics, 2 Academician Glushkov Avenue,
031127 Kyiv, Ukraine
2 Astrakhan State University, 20a Tatishchev St., 414056 Astrakhan, Russia
3 Institute of Biocolloidal Chemistry named after F.D. Ovcharenko of the National Academy of Sciences of
Ukraine, 42 Academician Vernadsky Boulevard, 03142 Kiev, Ukraine
Received May 8, 2014
This work studies the jamming and percolation of parallel squares in a single-cluster growth model. The Leath-
Alexandrowicz method was used to grow a cluster from an active seed site. The sites of a square lattice were
occupied by addition of the equal size k ×k squares (E-problem) or a mixture of k ×k and m ×m (m É k)
squares (M-problem). The larger k ×k squares were assumed to be active (conductive) and the smaller m ×m
squares were assumed to be blocked (non-conductive). For equal size k ×k squares (E-problem) the value of
p j = 0.638±0.001 was obtained for the jamming concentration in the limit of k →∞. This value was noticeably
larger than that previously reported for a random sequential adsorption model, p j = 0.564± 0.002. It was
observed that the value of percolation threshold pc (i.e., the ratio of the area of active k ×k squares and the
total area of k ×k squares in the percolation point) increased with an increase of k. For mixture of k ×k and
m×m squares (M-problem), the value of pc noticeably increased with an increase of k at a fixed value of m and
approached 1 at k Ê 10m. This reflects that percolation of larger active squares in M-problem can be effectively
suppressed in the presence of smaller blocked squares.
Key words: jamming, percolation, squares, disordered systems, Monte Carlo methods, Leath-Alexandrowicz
method
PACS: 02.70.Uu, 05.65.+b, 36.40.Mr, 61.46.Bc, 64.60.ah
1. Introduction
Percolation and jamming problems for extended objects of various shapes and sizes, deposited on the
lattices in two dimensions (2d), continuously attract great interest [1–3]. The model of random sequential
adsorption (RSA) is frequently used to form a random deposit on a substrate. In RSA model, the newly
placed particle cannot overlap with the previously deposited ones, the adsorbed objects remain fixed and
the final state is a disordered one (known as the jamming state) [4]. The fraction of the total surface, occu-
pied by the adsorbed particles, is called the jamming concentration, p j . The objects with different shapes
and sizes (e.g., linear [5–12] and flexible (polymer-like) [13, 14] k-mers (particles occupying k adjacent
sites), T-shaped objects and crosses [15], squares [16–18], disks [19], ellipses [4]) have been studied, and
data of these studies show that the value of p j strongly depends on the object shape and size.
The square-shaped particles on planar substrates have been studied in numerous works as useful
objects for a description of both fundamental and practical problems. The squares have been used as a
model of anisotropic 2d “molecules” in equilibrium systems using different theoretical and Monte Carlo
approaches [20–23]. The highest density for square particles is 1, however, even in that state the particles
can be arranged into uncorrelated parallel rows with non-crystalline order. Parallel squares appear to
∗This work is dedicated to the memory of Professor Alexander I. Olemskoi.
†E-mail: lebovka@gmail.com
© I.A. Kriuchevskyi, L.A. Bulavin, Yu.Yu. Tarasevich, N.I. Lebovka, 2014 33006-1
http://dx.doi.org/10.5488/CMP.17.33006
http://www.icmp.lviv.ua/journal
I.A. Kriuchevskyi et al.
exhibit a second-order melting transition (with the pressure being continuous and the compressibility
discontinuous) at a concentration p of 0.79 [23]. Square-shaped particles are also interesting as potential
mesogens. Monte Carlo simulations of squares have found a tetratic intermediate phase with the quasi-
long-range orientational order and the translational order decaying faster than algebraically [24]. The
rapid change of the orientational order of squares was observed at coverage of ≈ 0.69.
Different variants of a non-equilibrium RSA model for squares deposited on 2d substrates were also
developed. The coordination RSA model for k ×k squares (squares of side k) deposited on a square lat-
tice was studied numerically [25]. In this model, the squares are not allowed to touch one another if the
number of contact exceeds the predefined value of average coordination number c. It was shown that
the jamming coverage is ≈ 0.56 in the limit of k → ∞, independent of c. Moreover, for an average co-
ordination of about 2.4, the jamming coverage was ≈ 0.58, independent of the size k. The effect of size
distribution on the jamming coverage of parallel squares on the substrate was recently studied [26]. A
power distribution of square sizes can lead to a much larger packing density than for the equal size
squares.
The numerical simulations of RSA deposition of k ×k squares on a square lattice gave the following
power dependence of jamming concentration p j (k) [27, 28]:
|p j −p∞
j |∝ k−α
, (1.1)
where p∞
j
= 0.564± 0.002 corresponds to the continuous limit (k → ∞) and α ≃ 1. Thus, the jamming
concentration is in inverse proportion to the size of the square, k.
However, the percolation for the RSA model of k ×k squares was observed only at small values of k
(k = 1−3). However, only finite clusters of k ×k squares were observed at saturation coverage for k Ê 4.
The values of jamming p j and percolation pc concentrations at different sizes of squares and different
types of packing on the substrate are presented in table 1.
Table 1. Jamming p j and percolation pc concentrations for RSA packing of squares in square lattice (SL)
and continuous (C) systems.
Model System p j pc Reference
1×1, RSA, NNa SL 0.36413 − [18]
1×1, RSA, NNNb SL 0.18698 − [18]
1×1, RSA SL 1 0.592746 [29]
2×2, RSA SL 0.74788 0.601 [18]
3×3, RSA SL 0.681 0.621 [16]
4×4, RSA SL 0.646 − [16]
k ×k , RSA SL 0.564 − [16]
O, RSAc C 0.562009 − [30]
O, RLPd C 0.75 − [31]
O, RSAe C 0.327 − [26]
RO, RSAf C 0.532 − [17]
aNearest-neighbor exclusion
bNearest- and next-nearest-neighbour exclusion
cOriented
dRandom loose packing
eEach square has a single chance of adsorption
fRandomly oriented
The percolation in the mixtures of squares with different sizes was also intensively studied [32–38].
The percolation k×k squares at k Ê 4 can be restored by adding supplementary (k−1)×(k−1) squares of
smaller size to the jammed system. The calculations have shown that percolation threshold is pc ≈ 0.73
at large k (k Ê 15) [32, 33]. The study of percolation in the mixtures of monomers (k = 1) with 2×2 and
4× 4 squares has shown [34, 35] that percolation threshold has increased compared with its value for
33006-2
Jamming and percolation of parallel squares
the ordinary percolation of monomers, pc = 0.592746 [29]. E.g., for the mixture of monomers (k = 1) and
2× 2 squares at equal fractions of the total area, occupied by 1× 1 and 2× 2 squares, the percolation
concentration was 0.715±0.05 [34, 35]. The model of composite containing monomers (conductors) and
k×k squares (insulators) that fill the space in a regular manner was shown to be useful in explaining the
experimental data on percolation in segregated polymers [36].
The present work is devoted to the study of jamming and percolation of the equal size k×k squares (E-
problem) and their mixtures with smaller m ×m squares (M-problem) in a single-cluster growth model.
The Leath–Alexandrowicz (LA) method was used to grow a cluster from an active seed site [39, 40]. LA
method was intensively used to study the ordinary percolation problem for monomers. The deposition
rules in a single cluster model are obviously different from those in RSA model and it is expected that
percolation and jamming behaviors in single cluster and RSA models should be quite different. The re-
mainder of the paper is organized as follows. In section 2, we describe our model and the details of
simulation. The obtained results are discussed in section 3. We summarize the results and conclude our
paper in section 4.
2. Description of models and details of simulations
The Leath-Alexandrowicz (LA) method [39, 40] was used to grow a cluster from an active monomer
seed on the square lattice. The lattice sites were occupied by addition of the equal size k ×k squares (E-
problem) or a mixture of k ×k and m ×m (m É k) squares (M-problem). In M-problem, k ×k and m ×m
squares were assumed to be active (conductive) and blocked (non-conductive), respectively. Note that the
simplest case of m = 1 corresponds to the mixture of k ×k squares and monomer. In order to grow the
cluster of equal size k ×k squares (E-problem), LA algorithm uses the following steps (see figure 1):
1. Occupy an initial seed by a monomer. It has 4 initial perimeter sites;
2. Deposit randomly the first k ×k square attached to this seed. Determine new perimeter sites;
3. Randomly choose one perimeter site and try to fill the lattice sites with a new k ×k square. This
can be done by two equiprobable ways in horizontal or vertical directions (figure 1). In the case of
unsuccessful attempt, continue step 3;
4. Denominate the new added k ×k square as active with probability ρ and as blocked with proba-
bility 1−ρ. Add an active k ×k square to the cluster and determine new perimeter sites. The sites,
occupied by the blocked k ×k square, are not tested again;
5. Continue steps 3 and 4 until there remain no untested perimeter sites.
5x5 square
Selected site
on a perimeter
Two different positions
monomer
seed
seed
horisontal vertical
Direction
Figure 1. (Color online) To the description of the computation LA algorithm to grow clusters of k × k
squares (E-problem).
33006-3
I.A. Kriuchevskyi et al.
The time of grown t was evaluated as the number of deposited k ×k squares.
The similar LA algorithm was used for the M-problem [for mixtures of k × k and m × m squares
(m É k)]. The relative fraction of active k ×k squares (i.e., the ratio of the area occupied by k ×k squares
and the total area occupied by k ×k and m ×m squares) was calculated as follows:
p =
ρk2
ρk2 + (1−ρ)m2
=
[
1− (1/ρ−1)m2
/k2
]−1
. (2.1)
Note, that for the E-problem, m = k and p = ρ is the fraction of active squares.
Figure 2. (Color online) Examples of clusters for E-problem [equal size k ×k squares, p = 0 (a) and p , 0
(b)] and M-problem [a mixture of k ×k and m ×m (m É k) squares (c)]. Here, gray squares are active
(conductive) and black squares are blocked (non-conductive).
Examples of clusters of k×k squares for E-problem are presented in figure 2 (a) (p = 0) and figure 2 (b)
(p , 0). For Monte Carlo simulations with p = 1, the deposition was terminated when the jamming state
was reached, and these data were used to calculate the jamming concentration p j . The percolation con-
centration pc was estimated as the threshold concentration that divides the regimes of infinite and finite
clusters growth. At p < pc, only finite clusters were grown.
Example of a cluster of k ×k squares, blocked by smaller m ×m squares, is presented in figure 2 (c).
The E- and M-problems with k = 2−64 and m = 1,2,4,8 on a square lattice of L ×L size have been
studied. The random number generator of Marsaglia et al. was used in these studies [41]. The numerical
data of Monte Carlo simulations were analyzed for different simulations by the finite-size scaling of a
linear lattice with size L varied from 128 to 8192. The data were averaged using 1000 independent runs
for L É 2048 and 100–500 runs for larger systems.
3. Results and discussion
3.1. Equal sized k ×k squares (E-problem)
Figure 3 presents examples of the finite scaling analysis of jamming concentrations p j at different
values of k. At large size of the lattice (L > 20k), the observed dependencies between p j and 1/L were
practically linear (see, inset to figure 3) which was similar to the observation for RSA problem of k ×k
squares on a square lattice [27, 28]. To demonstrate the finite-size effects on the jamming concentration
more clearly, the results are represented in the form of p j (L)−p j (L →∞) versus the inverse lattice size
1/L.
Figure 4 shows the jamming concentration in the thermodynamic limit (L →∞), hereinafter referred
to as p j , versus the size of the square, k. It has been found that numerical results may be rather well
fitted by a power law function, equation (1.1), with parameters p∞
j
= 0.638±0.001 and α= 1.053±0.002.
The inset to figure 4 presents p j −p∞
j
versus k dependencies for single cluster and RSAmodels, where
the p j (k) data for RSAmodel were extracted from [27, 28]. Here, the dashed line corresponds to the slope
33006-4
Jamming and percolation of parallel squares
0 0.0005 0.001 0.0015 0.002
0
0.005
2
4
8
32
0 0.05 0.1
0.7
0.75
0.8
1/L
1/L
p j
-p
j(
L
→
∞
)
p j
k=
)( ∞→Lp j
Figure 3. (Color online) Examples of the finite scaling analysis of jamming concentrations p j at different
sizes of the square k. Here, p j (L →∞) is the thermodynamic limit (L →∞) of the jamming concentration
p j for the single-cluster growth model of equal-sized k ×k squares (E-problem). The inset presents p j
versus 1/L dependencies for 1/L É 0.1. Error bars are smaller than the symbols.
20 40 60 80 100
0.6
0.7
0.8
0.9
1
100 101
10-2
10-1
k
k
p j
p j
-
p j
∞
001.0638.0 ±≈∞
jp
002.0564.0 ±≈∞
jp
LA model
LA model
RSA model
RSA model
Figure 4. (Color online) The jamming concentration in the thermodynamic limit (L →∞), p j , versus the
size of the square, k. The data are presented for the single-cluster growth model (E-problem) and RSA
model (the data extracted from [27, 28] of deposition of the equal-sized k ×k squares). The inset presents
p j −p∞
j
versus k dependencies for single cluster and RSA models. Here, p∞
j
is the limiting value of p j at
k →∞. The dashed line corresponds to the slope of −1. Error bars are smaller than the symbols.
33006-5
I.A. Kriuchevskyi et al.
0.45 0.5 0.55 0.6 0.65
0
0.2
0.4
0.6
0.8
1
512
1024
2048
4096
0 0.005
0.56
0.58
0.6
0.62
2
4
12
32
L=
k=16
k=
p
L-1/ν
R
p c
)( ∞→Lpc
Figure 5. (Color online) Examples of the percolation probability R versus the fraction of active squares p
for the single-cluster growth model of equal size k ×k squares (E-problem, k = 16) and different values
of L. The inset shows pc versus L−1/ν dependencies for different values of k. Here, ν= 4/3 is the critical
exponent of correlation length for the ordinary 2D random percolation problem [42]. Error bars are
smaller than the symbols.
of −1. This evidences that the jamming concentration p j is in inverse proportion to the size of the square
k for both the single-cluster growth and RSA models.
However, at the same values of k , the values of p j were larger for single-cluster model than for
RSAmodel. This reflects that single-cluster growth rules give more compact packing than RSA deposition
rules.
Figure 5 shows the examples of the percolation probability R versus the fraction of active squares
p for the single-cluster growth model of equal size k × k squares (k = 16) and different values of L.
To extrapolate estimations of the percolation thresholds pc(L), obtained at the lattice of size L, to the
infinitely large lattice pc, the usual finite-size scaling analysis of the percolation behavior was done. To
perform extrapolation, the scaling relation
∣
∣pc −p∞
c
∣
∣∝ k−1/ν
, (3.1)
was used.
Here, ν = 4/3 is the critical exponent of correlation length for the ordinary 2D random percolation
problem [42]. In our study, the typical values of lattice size were L É 100k. The inset to figure 5 shows
pc versus L−1/ν dependencies for different values of k. Our results evidence that the problem studied
belongs to universality of the ordinary 2D random percolation problem at different values of k [42].
Figure 6 presents the percolation threshold pc versus the size of a square k for the single-cluster
growth model of equal size k ×k squares. The value of pc continually increased with an increase of k
in the studied range of k = 2. . . 64. In the problem under consideration, the formation of the percolation
cluster reflects the connectivity between the central seed and infinitely distant k ×k squares through the
network of active k×k squares, and each blocked k×k square terminates the growth of the cluster in the
vicinity of this square. It may be assumed that this connectivity is similar to that in the Bethe lattice in the
limit of k →∞. The percolation threshold for the Bethe lattice can be calculated as pc = 1/(z −1) where
z is the coordination number. In the limit of k →∞, the studied E-problem corresponds to a continuous
(off-lattice) problem with the maximum coordination number z = 4. Thus, it may be speculated that for
our E-problem, p∞
c ≈ 0.75.
33006-6
Jamming and percolation of parallel squares
100 101 1020.59
0.6
0.61
0.62
0.63
p c
k
Figure 6. (Color online) Fraction of active squares at the percolation point pc versus the size of a square
k for the single-cluster growth model of equal size k ×k squares (E-problem).
Dashed line in figure 6 corresponds to the following power relation:
pc = 0.75−ak−α
, (3.2)
where a = 0.156±0.001 and α= 0.051±0.001.
We see that computational data points can be satisfactorily fitted using equation (3.2). However,
larger scale computations are required in future in order to make a more precise estimation of the value.
3.2. Mixture of k ×k and m ×m (m É k) squares (M-problem)
Figure 7 presents examples of the percolation probability R versus the fraction of active k×k squares
p for the mixture of k × k squares and monomers. Figure 7 (a) compares R(p) dependencies for the
single-cluster LA growth model (M-problem) and RSA model (ρ = 0.95) (a) for k = 2. In RSA model, the
inactive (insulating) monomers were initially deposited on the square lattice with probability of 1−ρ.
Then, k ×k squares were deposited using RSA algorithm and the relative fraction of active k ×k squares
was calculated using equation (2.1). It is interesting that the introduction of blocking inactive (insulating)
monomers resulted in a noticeable increase of threshold concentration pc. E.g., for k = 2 in absence
of blocking monomers, the value of pc was ≈ 0.600−0.601 both for single cluster LA and RSA models.
However, in presence of blocking monomers, the value of pc increased up to 0.782 and 0.993, for LA and
RSA models, respectively.
Figure 7 (b) compares R(p) dependencies for M-problem for different values of k. These data were
used for a finite scaling analysis and determination of the percolation threshold, pc (figure 8).
The results show that for the mixture of k ×k and m ×m squares, the percolation threshold pc was
increasing noticeably with an increase of k at a fixed value of m (figure 8). Moreover, it approached 1 at
k Ê 10m. Thus, for the model of the mixture of k ×k and m ×m squares, the percolation of larger active
k ×k squares may be suppressed in the presence of considerably smaller blocked m ×m squares. This is
a natural account for the effective blocking of the side faces of k ×k squares at k ≫ m.
Finally, note that detailed experiments revealed the difference between the actual, pf, and predefined,
p , fractions of active squares for M-problem (a mixture of k ×k and m ×m squares). The Monte Carlo
simulation evidences that the actual value pf was noticeably smaller than the pre-determined p value
33006-7
I.A. Kriuchevskyi et al.
0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
L=
- 256
- 512
- 1024
- 2048
- 4096
p
R 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
L=
- 64
- 128
- 256
k=2 4 8
k=2 k=2
RSA modelLA model
LA model
a)
b)
Figure 7. (Color online) Examples of the percolation probability R versus the fraction of active k × k
squares p for mixture of k ×k squares and monomers. Here, the R(p) dependencies are compared for
the single-cluster LA growth model (M-problem) and RSA model (ρ = 0.95) (a) for k = 2 (a) and for LA
growth model (M-problem), k = 2,4,8 (b). The data are presented for different values of L. Error bars are
smaller than the symbols.The positions of the percolation threshold are shown by arrows.
100 101 102
0.6
0.7
0.8
0.9
1
m=1 2 4
k
p c
E-problem
M-problem
Figure 8. (Color online) Fraction of active k ×k squares at the percolation point pc versus the size of a
square k for E-problem (equal size k ×k squares) and M-problem (a mixture of k ×k and m ×m (m É k)
squares). Error bars are smaller than the symbols.
(figure 9). This fact may be explained as follows. In the course of the cluster growth, the network of finite
size pores was formed. We can expect that the probability of deposition is smaller for k ×k squares due
to stronger spatial restrictions and possibility of rejection of the deposition attempts. This results in the
reduction of the actual value of pf in the course of the growth [figure 9 (a)]. This effect is quite similar to
the observed differences between pre-determined and actually observed orientation order parameters
in RSA model for partially oriented k-mers on a square lattice [10, 11]. For example, figure 9 (b) demon-
33006-8
Jamming and percolation of parallel squares
0 2000 4000 6000 8000 10000
0.7
0.8
0.9
1
0.7 0.75 0.8 0.85 0.9 0.95 1
0.7
0.75
0.8
0.85
0.9
0.95
1
2
3
4
8
16
t
p
p f
p f k=
0.80
0.85
0.90
0.95
0.99
p=a)
b)
Figure 9. (Color online) Actual value of the fraction of active squares pf versus the time of growth t at
different predefined values of p (a) and values of pf versus p at different values of k. Data are for a
mixture of k×k squares and monomers above the percolation threshold of active squares, i.e., at p Ê pc.
strates that this difference between pf and p may be rather noticeable for a mixture of k ×k squares and
monomers above the percolation threshold of active squares and it was growing with an increase of the
square size k.
4. Conclusion
In this paper, the jamming and percolation of parallel squares in a single-cluster growth model were
investigated by computer simulations. The sites of a square lattice were occupied by addition of equal
size k ×k squares (E-problem) or a mixture of k ×k and m ×m (m É k) squares (M-problem). The larger
k ×k squares were assumed to be active (conductive) and the smaller m ×m squares were assumed to
be blocked (non-conductive). For jamming concentration of equal size squares (E-problem), the power
low dependence of type | p j −p∞
j
|∝ k−α was obtained, where p∞
j
= 0.638±0.001 and α≈ 1.0. The data
also evidence that the studied problem belongs to the universality of ordinary 2D random percolation at
different values of k. The percolation threshold pc increased with an increase of k. It was speculated that
pc(k) can be described by the relation pc = 0.75− ak−α , where a = 0.156±0.001 and α = 0.051±0.001.
For mixture of k × k and m ×m (m É k) squares (M-problem), the percolation threshold pc increased
noticeably with an increase of k at fixed value of m and approached 1 at k Ê 10m. It was demonstrated
that percolation of larger active k ×k squares can be effectively suppressed in the presence of smaller
blocked m ×m squares for the M-problem.
5. Acknowledgements
Authors would like to acknowledge the partial financial support of project 43–02–14(U), Ukraine (N.L.)
and of project RFBR 14–02–90402_Ukr, Russia (Yu.T.). Authors also thank Dr. N.S. Pivovarova for her help
with the preparation of the manuscript.
33006-9
I.A. Kriuchevskyi et al.
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Jamming and percolation of parallel squares
Джамiнг та перколяцiя паралельних квадратiв в
однокластернiй моделi росту
I.О. Крючевський1, Л.А. Булавiн1, Ю.Ю. Тарасевич 2, М.I. Лебовка3
1 Київський нацiональний унiверститет iм. Тараса Шевченка, фiзичний факультет,
пр. академiка Глушкова, 2, 03127 Київ, Україна,
2 Астраханський державний унiверситет, вул. Татiщева, 20a, 414056 Астрахань, Росiя
3 Iнститут бiоколоїдної хiмiї iм. Ф.Д. Овчаренка НАН України,
бульв. академiка Вернадського, 42, 03142 Київ, Україна
В роботi вивчено явища джамiнгу i перколяцiї паралельних квадратiв для однокластерної моделi росту.
Для росту кластеру з активного зародку використовувався метод Лiса-Александровича. Вузли квадратної
ґратки займалися додаванням однакових k ×k квадратiв (E-задача) або сумiшi k ×k i m ×m (m É k) ква-
дратiв (M-задача). Припускалося, що бiльшi k × k областi були активними (провiдними), а меншi були
заблокованими (непровiдними). Для k ×k квадратiв однакового розмiру (E-задача) за умови k →∞ було
отримано таке значення концентрацiї джамiнгу p j = 0.638±0.001 . Це значення було iстотно меншим за
отримане ранiше для моделi випадкової послiдовної адсорбцiї: p j = 0.564±0.002. Було показано, що ве-
личина перколяцiйного порогу pc (тобто вiдношення площi активних k ×k квадратiв до загальної площi
осаджених k ×k квадратiв в перколяцiйнiй точцi) зростала при збiльшеннi k. Для сумiшi k ×k i m ×m
квадратiв (M-задача) величина pc сильно зростала при збiльшеннi k при фiксованому значеннi m та на-
ближалась до 1 приk Ê 10m. Це пов’язано з тим, що перколяцiя бiльших активних квадратiв для M-задачi
може ефективно пригнiчуватися за наявностi невеликої кiлькостi малих заблокованих квадратiв.
Ключовi слова: джамiнг, перколяцiя, квадрати, невпорядкованi системи, метод Монте Карло,
метод Лiса-Александровича
33006-11
Introduction
Description of models and details of simulations
Results and discussion
Equal sized k k squares (E-problem)
Mixture of k k and m m (mk) squares (M-problem)
Conclusion
Acknowledgements
|