ArticlePDF Available

Abstract and Figures

In this paper we apply lattice models of finite binary percolation networks to examine the effects of network configuration on macroscopic network responses. We consider both square and rectangular lattice structures in which bonds between nodes are randomly assigned to be either resistors or capacitors. Results show that for given network geometries, the overall normalised frequency-dependent electrical conductivities for different capacitor proportions are found to converge at a characteristic frequency. Networks with sufficiently large size tend to share the same convergence point uninfluenced by the boundary and electrode conditions, can be then regarded as homogeneous media. For these networks , the span of the emergent scaling region is found to be primarily determined by the smaller network dimension (width or length). This study identifies the applicability of power-law scaling in random two phase systems of different topological configurations. This understanding has implications in the design and testing of disordered systems in diverse applications.
Content may be subject to copyright.
RESEARCH ARTICLE
Universality of the emergent scaling in finite
random binary percolation networks
Chongpu Zhai
1
, Dorian Hanaor
1,2
, Yixiang Gan
1
*
1The School of Civil Engineering, The University of Sydney, Sydney, New South Wales, Australia, 2Institute
for Materials Science and Technology, Technische Universita¨t Berlin, Berlin, Germany
These authors contributed equally to this work.
*yixang.gan@sydney.edu.au
Abstract
In this paper we apply lattice models of finite binary percolation networks to examine the
effects of network configuration on macroscopic network responses. We consider both
square and rectangular lattice structures in which bonds between nodes are randomly
assigned to be either resistors or capacitors. Results show that for given network geome-
tries, the overall normalised frequency-dependent electrical conductivities for different
capacitor proportions are found to converge at a characteristic frequency. Networks with
sufficiently large size tend to share the same convergence point uninfluenced by the bound-
ary and electrode conditions, can be then regarded as homogeneous media. For these net-
works, the span of the emergent scaling region is found to be primarily determined by the
smaller network dimension (width or length). This study identifies the applicability of power-
law scaling in random two phase systems of different topological configurations. This under-
standing has implications in the design and testing of disordered systems in diverse
applications.
Introduction
The bulk behavior of complex systems comprising disordered multi-phase components is of
importance in diverse applications including supercapacitors and batteries [14], dielectric
material characterization [510], the mechanics of structures [1113], fracturing process [14],
thermal analysis [15] and soil probing [16]. In such systems, various parameters govern the
electrical, thermal, chemical, and/or mechanical properties of components of a system across
multiple scales from molecular up to macroscopic length-scale. Experimental and computa-
tional research efforts are increasingly conducted in order to gain insights into the manner in
which these properties combine across scales to determine overall system performance.
In particular, the AC conductivity of systems that can be schematically represented as mix-
tures of electrical components has been the subject of numerous investigations that have
shown power-law scaling with frequency arising through different relaxation mechanisms [11,
1719]. Above a critical frequency this scaling of AC conductivity is described by Jonscher’s
power law[20] and has been experimentally observed across diverse conductor-dielectric
PLOS ONE | DOI:10.1371/journal.pone.0172298 February 16, 2017 1 / 11
a1111111111
a1111111111
a1111111111
a1111111111
a1111111111
OPEN ACCESS
Citation: Zhai C, Hanaor D, Gan Y (2017)
Universality of the emergent scaling in finite
random binary percolation networks. PLoS ONE 12
(2): e0172298. doi:10.1371/journal.pone.0172298
Editor: Dante R. Chialvo, Consejo Nacional de
Investigaciones Cientificas y Tecnicas, ARGENTINA
Received: November 22, 2016
Accepted: February 2, 2017
Published: February 16, 2017
Copyright: ©2017 Zhai et al. This is an open
access article distributed under the terms of the
Creative Commons Attribution License, which
permits unrestricted use, distribution, and
reproduction in any medium, provided the original
author and source are credited.
Data Availability Statement: All relevant data are
included within the paper.
Funding: Financial support for this research is from
the Australian Research Council through grants
DE130101639 is greatly appreciated. The funder
had no role in study design, data collection and
analysis, decision to publish, or preparation of the
manuscript.
Competing interests: The authors have declared
that no competing interests exist.
composites and porous materials [5,7,18,21,22] with this scaling being termed the “Universal
Dielectric Response” [17,18,21,23]. This emergent property does not arise directly from any
particular physical or chemical properties of the involved components, but rather is a conse-
quence of the way components combine [11,19,21,24,25]. Such dielectric mixtures have
been effectively approximated as a random network of resistors and capacitors [18,19,24]
with representative conductors exhibiting a constant conductance 1/Rand dielectric compo-
nents exhibiting a variable complex admittance iωC, which is directily proportional to an
angular frequency ω, as illustrated in Fig 1. Useful asymptotic formula for the emergent net-
work admittance including both the effects of component proportions and the network size
can be obtained based on the spectral method [21,26] and the averaging approach [11,21,27].
However, establishing a more rigorous estimation necessitates numerical analysis.
From previous numerical studies [11,1719,21,23,25,28], the typical obtained conductiv-
ity-frequency spectrum of a square lattice resistor-capacitor (RC) network can be divided into
three regions of angular frequency, ω, governed by the proportion of capacitors, p
c
, and net-
work size, N(a) an emergent region for intermediate frequencies; (b) two percolation regions;
(c) transitions between the two above-mentioned [21]. Symmetry is found between the overall
responses at high and low frequencies, which can be correlated to percolation behavior [26,29,
30]. For low and high frequencies, current tends to percolate predominantly through resistors
and capacitors respectively, as these will exhibit relatively lower impedance, as presented in Fig
1. For intermediate frequencies, where the values of admittance for the resistors and the
Fig 1. A lattice network containing W×L = 5×5 randomly distributed resistors and capacitors. The network width Wand
length Lindicate the numbers of horizontal and vertical elements in a single chain, respectively. The values of B
0
= 2, B
1
= 4, and
B
2
= 6 present the number of nodes connected directly to the electrodes. Percolation paths formed by resistors and capacitors are
shown in thick blue and orange lines, corresponding respectively to the dominating modes at low and high frequencies, for the B
0
configuration.
doi:10.1371/journal.pone.0172298.g001
Emergent scaling in binary percolation networks
PLOS ONE | DOI:10.1371/journal.pone.0172298 February 16, 2017 2 / 11
capacitors are close, we observe the power-law emergent behavior whereby conductivity is
proportional to ω
α
, with αp[11,21,27].
Motivated by the applicability of these networks for representing real-world disordered sys-
tems, in this paper we reexamine the universality of the emergent power-law scaling observed
in previous work by further considering the significance of the network aspect ratio and
boundary conditions on the convergence point and the span of the emergent region, the two
key network characteristics of universal scaling behavior.
Methods
In this paper, we extend the square lattice RC networks to rectangular ones with N=W×Lele-
ments distributed between two bus-bars, one of which is grounded and the other raised to a
potential, |V
0
|e
iωt
. A system of complex number linear equations is set up by applying Kirchh-
off’s current law (for complex currents) on each individual node of the RC network. For the
node k, we get
IkðtÞ ¼ Sn
jjIj;kjeiðotþφj;kÞ¼Sn
jðjVj;kjeiðotþφj;kÞ jVkjeiðotþφkÞÞ=Zj;k¼0;ð1Þ
where I
k
is the sum of the currents (negative or possitive) flowing towards the node k, from
connected components. The impedance of a component connected to node k,Z
j,k
, is random-
ized to be either Ror 1/iωC. The voltage potential of the connected node j,jVj;kjeiðotþφj;kÞ, with
respect to that of the node k, is represented as complex-valued function of time, t, with φ
j,k
being the relative phase. The value of nis determined by the location in the network, equaling
the number of connected components. More specifically, n= 4 for ones located away from the
boundaries, n= 2 for the lattice corners, n= 3 for the nodes on the boundaries excluding cor-
ners. The two electrodes are also regarded as nodes with n=B. The electrode dimension, B, is
defined as the number of nodes connected directly to the electrodes, e.g., B=W+ 1 when all
the elements along the boundary side are connected to the electrode. Each single node is repre-
sented by a corresponding linear algebraic equation, resulting in (W+ 1) ×(L+ 1) equations
for all the nodes. Two additional equations can be obtained from the electrodes. By solving
these equations, the potential of each node and the current going through each bonds in the
network can be calculated. Thus, for a given applied potential difference between electrodes,
|V
0
|e
iωt
, the macroscopic admittance can be given by Y¼ jI0jeiðotþφIÞ=jV0jeiot, where jI0jeiðotþφIÞ
is the obtained overall current flowing into the ground. Additionally, Frank-Lobb techniques
can be employed to reduce the network size, thus improving the computational efficiency [31].
Here, the network aspect ratio and the electrode dimension are considered as variables, in
order to investigate the influence of network configuration and boundary conditions on mac-
roscopic responses. The network size is determined by its width, W, and length, L, which rep-
resent the numbers of components in a single chain along the horizontal and vertical
directions, respectively. In the network circuit, two electrodes of identical dimension are con-
nected to elements located symmetrically in the center of the two vertical boundaries. The fre-
quency-dependent macroscopic responses obtained from different configurations, in terms of
network length, L, and width, W, are normalized through
~
Y¼jYjRL
Wþ1;e
o¼oRC;ð2Þ
where Rand Care the resistance and capacitance values of resistors and capacitors in the con-
sidered network, respectively. This normalization process is applied in order to include the sig-
nificance of all the elements in the overall network behavior, represented by the equivalent
admittance, by considering rules for simple series and parallel combinations of components.
Emergent scaling in binary percolation networks
PLOS ONE | DOI:10.1371/journal.pone.0172298 February 16, 2017 3 / 11
The span of the emergent region, S, is defined as the horizontal distance between the intersec-
tions of the power-law function, y¼opc, with top and bottom percolation admittances, e
Y1
and e
Y2(averaged from multiple simulations), for low and high frequencies, respectively, as
shown in Fig 2.
The network behaviors shown in Fig 2 in the frequency domain are primarily governed by
percolation effects [32], which are closely linked to the frequency-dependent conductivity of
each single bond in the network. In the studied rectangular RC networks with two types of
bonds (i.e., resistor and capacitor, the admittance ratio of capacitor elements with respect to
resistors is iωRC) have been considered to describe the responses of random binary networks.
The observed universal scaling behavior in Fig 2 can be also found in other networks
Fig 2. Normalised admittance module as a function of frequency. Numerical results obtained from three groups of differently
configured networks (denoted as W×L_B) are presented, with capacitor proportions, p
c
, varying from 0% (corresponding curves are
shown in black) to, 25% (red), 50% (blue), 75% (green), and to 100% (brown). The phase responses are depicted in the inset. For
each network configuration with a given capacitor proportion, five simulations have been realized.
doi:10.1371/journal.pone.0172298.g002
Emergent scaling in binary percolation networks
PLOS ONE | DOI:10.1371/journal.pone.0172298 February 16, 2017 4 / 11
containing two types of elements, indexed by aand b, exhibiting differences which can be
described in the form of S
a
=ωS
b
, e.g., mechanical stiffness, thermal conductivity, and chemi-
cal reaction rate, etc. [11,14,25].
Results and analysis
The AC electrical responses of three groups of different sized networks (expressed in the form
of W×L_B, representing network width×length_electrode dimension: 20 ×20_21, 100 ×
20_101, and 20 ×100_21) with varying p
c
are plotted in Fig 2. The three types of regions
observed in the conductance spectroscopy of square networks can be also observed here in
rectangular ones. For a given network geometry, the obtained admittance spectroscopies for
various p
c
, are found to intersect at a convergence point with the characteristic frequency of
ω= 1/RC where resistors and capacitors contribute equally to the overall conduction. This
point also appears to be the center of the emergent region. The normalized characteristic
admittance (the values of ~
Yat the characteristic frequency) at the convergence point is close to
1. This indicates that the network at the characteristic frequency perform effectively as a
mono-element network, which has a phase angle reaching the extremum value, as is shown in
Fig 2. It is further evidenced that the normalized emergent regions of the three groups of net-
works coincide with each other presenting universal features. The normalized admittance in
this common emergent region appears to be uninfluenced by the network aspect ratio (length/
width) or the electrode dimension. However, differences can be found at percolation regions
along with the corresponding transition regions. Variation of the width or length can poten-
tially change the percolation thresholds which will determine the responses, following a resis-
tive-percolated (plateaued) or capacitive-percolated (upwards or downwards) trend,
corresponding to the low and high frequency ranges, respectively.
Statistical analysis of the normalised characteristic admittance for different-sized square
networks (from 5 ×5 to 600 ×600) with B=W+ 1 was conducted and the standard deviation
(STD) of the normalized characteristic admittance is presented in Fig 3. It is found that all val-
ues of normalized characteristic admittance obtained with various p
c
(averaging over ten simu-
lations) are in the range of (0.95, 1.05) for networks with more than 10 ×10 elements. The
variance tends to diminish as the network size increases. This can be explained by considering
boundary effects that relatively smaller networks have higher percentage of boundary elements
(connected to four other elements rather than six in the bulk region). Responses of larger net-
works perform with little influence from the boundary. For a given sized square network a
larger variation is found for cases of p= 1/2, as such conditions lead to an equal likelihood of
resistive-percolated and capacitive-percolated network responses at low and high frequencies.
Consequently, there are four possible qualitatively different types of response for any realiza-
tion of the system.[21] Different available responses potentially introduce dispersion and
uncertainty of the network behavior in both percolation and emergent regions, as can be seen
in Fig 3.
We consider 2D rectangular networks with various L/W and B/(W+ 1) ratios, in order to
study the effects of network size and electrode dimension on frequency dependent responses.
The variations of normalized characteristic admittance for rectangular networks (not shown)
are comparable to those of square networks. Here, the convergence-divergence behavior is
tested with three groups of rectangular networks, which have the fixed width, W, of 20, 50, or
100 elements, respectively. The results obtained from the three groups coincide with each
other, and typical results for L/W = 1.5, 0.8, 0.2 are shown in the Fig 4A. The contour of nor-
malized characteristic admittance shown in Fig 4B presents a clear trend approaching 1, as the
network length and the electrode dimension increase. For an RC network with a given size, a
Emergent scaling in binary percolation networks
PLOS ONE | DOI:10.1371/journal.pone.0172298 February 16, 2017 5 / 11
smaller value of electrode dimension tends to constrict the current to fewer paths at zones near
the electrodes, thus, effectively reducing the network length. However, this influence will
diminish as the network lengthens. Networks with large enough length tend to share the same
intersection point uninfluenced by the boundary condition. In this case, these networks can be
defined as homogeneous systems, with the responses in emergent region unaffected by the net-
work configuration and the electrode dimension. A smaller length/width ratio or electrode
dimension will usually lead to normalized characteristic admittance values smaller than 1.
Fig 3. Mapping of the standard deviation of normalised characteristic admittance values.For varying capacitor proportions
from 0.1 to 1.0, different-sized square networks (from 5 ×5 to 600 ×600) were considered. For a given network size and capacitor
proportion, ten RC networks were generated and used in the simulations to obtain the averaged normalised characteristic
admittance, represented by the black dot. The STD of these points are used for mapping with the colour indicating the STD values, as
detailed in the legend.
doi:10.1371/journal.pone.0172298.g003
Emergent scaling in binary percolation networks
PLOS ONE | DOI:10.1371/journal.pone.0172298 February 16, 2017 6 / 11
Fig 4. Influences of network aspect ratio and electrode dimension on the values of characteristic
admittance. (A) The dependence of characteristic admittance on the B/(W+ 1) ratio, for cases of different
widths, W, of 20, 50, and 100 but same L/W, including 0.2, 0.8, and 1.5 (corresponding to ,, and ,
respectively). (B) Mapping of characteristic admittance values obtained from different-sized networks with
varying L/Wand B/(W+ 1) ratios. The square network is marked by the black dot.
doi:10.1371/journal.pone.0172298.g004
Emergent scaling in binary percolation networks
PLOS ONE | DOI:10.1371/journal.pone.0172298 February 16, 2017 7 / 11
For cases of electrodes of various geometries on the boundary or embedded in RC net-
works, the influence of the electrodes is mainly induced by the elements connected directly to
the electrodes. As the network size increases, boundary elements and electrode-affected ele-
ments will have a decreasing percentage. Therefore, the electrode size effect along with the
boundary effect will be unobservable for a sufficiently large network. The trend can be found
in Fig 4B that the characteristic admittance values for an increasingly large network locate well
upwards and to the right from the red zone, asymptotically approaching 1. This universal char-
acteristic can be also extended for an infinite network, with the normalized intersection value
to be 1.
The power law dependence of the electrical responses on frequency is of a universal nature
for a wide range of complex RC networks. To further interpret this emergent behavior, we
investigate the spans of the power-law emergent region. The convergence point tends to be the
geometric center of all the emergent regions for various p
c
. By considering the region center
reported here combined with the span, S, the emergent scaling behavior region can be well
described.
It has been observed [21] that, for a square network with p
c
= 0.5, the span of the power-law
emergent region increases without bound as the network size increases. In this paper, using
the results from square networks as references, we compare the spans of emergent regions
obtained for various sized rectangular networks with different electrode dimensions, as p
c
var-
ies. We found that when the normalized characteristic admittance value is sufficiently close to
1, it is the smaller dimension, S
min
= min(W,L) that determines the span of emergent region,
while the electrode dimension has little influence on the length. Different sized homogeneous
networks with the same s
min
tend to effectively present identical responses for the emergent
region with a given p
c
. However, discrepancies can be found for responses at low and high fre-
quencies dominated by percolation behavior. This likely to be the case also for an intersection
value far away from 1, but with lower accuracy due to the instability of the responses on
account of the boundary effects. Only the results with p
c
from 0 to 0.5 are discussed and pre-
sented in Fig 5, with the consideration of symmetricity of network responses.
The results shown here for varied network and electrode dimensions shed light on the
behavior of infinitely-large binary percolation-type network and large networks with irregular
boundaries (e.g., in the shape of spline curves) and electrodes (e.g., with the geometry of circu-
lar zones, embedded in the network, or unequal-sized electrodes). As long as the network size
is significantly larger than the electrode and boundary dimension, the presenting universal fea-
tures will not be influenced by the boundary and electrode conditions. This enables the net-
work responses to reach a robust and reliable status at the emergent region with the span being
determined by the effective network size in the order of De2(D
e
is the distance between positive
electrode and ground, indicating the shortest current path) and p
c
(dominating the slope of
universal power law). Additionally, evident trend presented in Fig 5 supports that infinitely
large emergent scaling regions can be observed for various capacitor proportions.
Conclusion
We studied the influences of network geometry and electrode dimension on the electrical
responses of rectangular RC networks. The universal scaling behavior can be fully character-
ized using the center and the length of the emergent region, i.e., the convergence point, and
the span, respectively. For both square and rectangular networks, a convergence point is
observed at the characteristic frequency, ω= 1/RC, which usually appears to be the center of
the emergent region. At this characteristic frequency, the normalized admittance value |Y|RL/
(W+ 1) approaches 1 as the length-to-width ratio and electrode dimension increase. For a
Emergent scaling in binary percolation networks
PLOS ONE | DOI:10.1371/journal.pone.0172298 February 16, 2017 8 / 11
defined homogeneous network, the span of the emergent range is primarily determined by the
shorter dimension of width and length. These observations provide a unified description for
the emergent scaling properties of network responses for random two-phase systems with
varying topological configurations. The comprehensive understanding of this emergent scaling
can guide the design and testing of disordered systems in terms of determining testing condi-
tions (e.g., the shape, size, location, and spacing of the fixtures), boundary conditions, and sys-
tem dimensions.
Author Contributions
Conceptualization: CZ DH YG.
Data curation: CZ.
Fig 5. Dependence of the emergent region span on network size and electrode dimension. The results for square networks are
shown by solid lines with error bars obtained across ten simulations for each point. Results of rectangular networks (10 ×100_11,
100 ×10_101 and 10 ×1000_5) are presented by dashed lines. The inset compares the spans of different-sized square networks with
p
c
= 0.5 with those of rectangular networks with various network configurations.
doi:10.1371/journal.pone.0172298.g005
Emergent scaling in binary percolation networks
PLOS ONE | DOI:10.1371/journal.pone.0172298 February 16, 2017 9 / 11
Formal analysis: CZ DH YG.
Funding acquisition: YG.
Investigation: CZ DH YG.
Methodology: CZ DH YG.
Project administration: YG.
Resources: YG.
Software: CZ.
Supervision: DH YG.
Validation: CZ YG.
Visualization: CZ YG.
Writing – original draft: CZ DH YG.
Writing – review & editing: CZ DH YG.
References
1. Bertei A, Choi H-W, Pharoah J, Nicolella C. Percolating behavior of sintered random packings of
spheres. Powder technology. 2012; 231:44–53.
2. Vo
¨lker B, McMeeking RM. Impact of particle size ratio and volume fraction on effective material parame-
ters and performance in solid oxide fuel cell electrodes. Journal of Power Sources. 2012; 215:199–215.
3. Zhai C, Hanaor D, Proust G, Brassart L, Gan Y. Interfacial electro-mechanical behaviour at rough sur-
faces. Extreme Mechanics Letters. 2016; 9:422–429.
4. Sabatier J, Aoun M, Oustaloup A, Gre
´goire G, Ragot F, Roy P. Fractional system identification for lead
acid battery state of charge estimation. Signal processing. 2006; 86(10):2645–57.
5. Papathanassiou A, Sakellis I, Grammatikakis J. Universal frequency-dependent ac conductivity of con-
ducting polymer networks. Applied Physics Letters. 2007; 91(12):122911.
6. Dyre JC, Maass P, Roling B, Sidebottom DL. Fundamental questions relating to ion conduction in disor-
dered solids. Reports on Progress in Physics. 2009; 72(4):046501.
7. Bakkali H, Dominguez M, Batlle X, Labarta A. Universality of the electrical transport in granular metals.
Scientific Reports. 2016; 6.
8. Li W, Schwartz RW. ac conductivity relaxation processes in CaCu3Ti4O12 ceramics: Grain boundary
and domain boundary effects. Applied physics letters. 2006; 89(24):242906.
9. Unuma T, Umemoto A, Kishida H. Anisotropic terahertz complex conductivities in oriented polythio-
phene films. Applied Physics Letters. 2013; 103(21):213305.
10. Nawroj AI, Swensen JP, Dollar AM. Electrically Conductive Bulk Composites through a Contact-Con-
nected Aggregate. PloS one. 2013; 8(12):e82260. doi: 10.1371/journal.pone.0082260 PMID: 24349239
11. Murphy K, Hunt G, Almond DP. Evidence of emergent scaling in mechanical systems. Philosophical
Magazine. 2006; 86(21–22):3325–38.
12. Picu R. Mechanics of random fiber networks—a review. Soft Matter. 2011; 7(15):6768–85.
13. Tighe BP. Dynamic critical response in damped random spring networks. Physical review letters. 2012;
109(16):168303. doi: 10.1103/PhysRevLett.109.168303 PMID: 23215140
14. Moreira A, Oliveira C, Hansen A, Arau
´jo N, Herrmann H, Andrade J Jr. Fracturing highly disordered
materials. Physical review letters. 2012; 109(25):255701. doi: 10.1103/PhysRevLett.109.255701
PMID: 23368480
15. Pollock H, Hammiche A. Micro-thermal analysis: techniques and applications. Journal of Physics D:
Applied Physics. 2001; 34(9):R23.
16. Deschamps R, Siddiqui S, Drnevich V. Time domain reflectometry development for use in geotechnical
engineering. 2000.
17. Almond D, Vainas B. The dielectric properties of random R-C networks as an explanation of theuniver-
sal’power law dielectric response of solids. Journal of Physics: Condensed Matter. 1999; 11(46):9081.
Emergent scaling in binary percolation networks
PLOS ONE | DOI:10.1371/journal.pone.0172298 February 16, 2017 10 / 11
18. Almond DP, Bowen C. Anomalous power law dispersions in ac conductivity and permittivity shown to
be characteristics of microstructural electrical networks. Physical review letters. 2004; 92(15):157601.
doi: 10.1103/PhysRevLett.92.157601 PMID: 15169318
19. Mccullen NJ, Almond DP, Budd CJ, Hunt GW. The robustness of the emergent scaling property of
random RC network models of complex materials. Journal of Physics D: Applied Physics. 2009; 42
(6):064001.
20. Jonscher AK. The universal dielectric response. nature. 1977; 267:673–9.
21. Almond D, Budd C, Freitag M, Hunt G, McCullen N, Smith N. The origin of power-law emergent scaling
in large binary networks. Physica A: Statistical Mechanics and its Applications. 2013; 392(4):1004–27.
22. Galvão RKH, Hadjiloucas S, Kienitz KH, Paiva HM, Afonso RJM. Fractional order modeling of large
three-dimensional RC networks. Circuits and Systems I: Regular Papers, IEEE Transactions on. 2013;
60(3):624–37.
23. Bouamrane R, Almond DP. The ‘emergent scaling’phenomenon and the dielectric properties of random
resistor–capacitor networks. Journal of Physics: Condensed Matter. 2003; 15(24):4089.
24. Creyssels M, Falcon E, Castaing B. Scaling of ac electrical conductivity of powders under compression.
Physical Review B. 2008; 77(7):075135.
25. Almond DP, Budd CJ, McCullen NJ. Emergent behaviour in large electrical networks. Approximation
Algorithms for Complex Systems: Springer; 2011. p. 3–26.
26. Jonckheere T, Luck J. Dielectric resonances of binary random networks. Journal of Physics A: Mathe-
matical and General. 1998; 31(16):3687.
27. Milton GW. Bounds on the complex dielectric constant of a composite material. Applied Physics Letters.
1980; 37:300–2.
28. Almond DP, Bowen C, Rees D. Composite dielectrics and conductors: simulation, characterization and
design. Journal of Physics D: Applied Physics. 2006; 39(7):1295.
29. Clerc J, Giraud G, Laugier J, Luck J. The electrical conductivity of binary disordered systems, percola-
tion clusters, fractals and related models. Advances in Physics. 1990; 39(3):191–309.
30. Bertei A, Nicolella C. A comparative study and an extended theory of percolation for random packings
of rigid spheres. Powder technology. 2011; 213(1):100–8.
31. Frank D, Lobb C. Highly efficient algorithm for percolative transport studies in two dimensions. Physical
Review B. 1988; 37(1):302.
32. Sakellis I, Papathanassiou A, Grammatikakis J. Scaling and universality of ac conductivity and dielectric
response in disordered materials under pressure. Applied Physics Letters. 2010; 97(4):042904.
Emergent scaling in binary percolation networks
PLOS ONE | DOI:10.1371/journal.pone.0172298 February 16, 2017 11 / 11
... , corresponding to representative values of a single interaction between a pair of spheres in the system, are similar [10,50], thus we have ω * = ...
... The obtained packing is sandwiched between two conductive rigid flats, one of which is grounded and the other is raised to a potential V (ω). The impedance can be obtained by solving the complex linear equations established by applying Kirchhoff's current law to each particle, denoting a node of the network [50]. A simulation volume larger than 10×10×10 particle diameters has been recommended to reduce the size dependence of the percolation threshold [54,55]. ...
... In order to investigate the role of packing structures in the universal scaling characterized by the master curve, we further applied the presented numerical framework with P c = 0.70 to networks with various configurations. Rather than 2D or 3D square lattice networks [16,19,22,50], we vary the cut-off distance defining the limits of particle interactions to obtain network structures with different coordination numbers [53,55,58] Fig. 5, a higher N c results in a later onset of conduction dispersion, lower resistive plateau value and sharper transition from resistive to NCL regimes. The coordination number indicating the connectivity of the network structure, is associated with the effective local dimension of current pathways, which fundamentally determines the conduction properties of network structures [58,59], and thus the transition from resistive plateau to dispersive conduction as frequency increases. ...
Article
We experimentally and numerically examine stress-dependent electrical transport in granular materials to elucidate the origins of their universal dielectric response. The ac responses of granular systems under varied compressive loadings consistently exhibit a transition from a resistive plateau at low frequencies to a state of nearly constant loss at high frequencies. By using characteristic frequencies corresponding to the onset of conductance dispersion and measured direct-current resistance as scaling parameters to normalize the measured impedance, results of the spectra under different stress states collapse onto a single master curve, revealing well-defined stress-independent universality. In order to model this electrical transport, a contact network is constructed on the basis of prescribed packing structures, which is then used to establish a resistor-capacitor network by considering interactions between individual particles. In this model the frequency-dependent network response meaningfully reproduces the experimentally observed master curve exhibited by granular materials under various normal stress levels indicating this universal scaling behaviour is found to be governed by i) interfacial properties between grains and ii) the network configuration. The findings suggest the necessity of considering contact morphologies and packing structures in modelling electrical responses using network-based approaches.
... The exponent of the emergent power-law scaling is directly related to capacitor proportion, P C in the established RC network, according to the mixing rule. 13-15 22 For a given RC network, the emergent power-law scaling of varying P C coincide at a universal intersection centre 17 at the frequency, f I , where resistors and capacitors contribute equally to the overall conduction i.e., 1/R = 2 f I C. Further with conduction information at low and high frequencies, the whole AC spectrum can be well described. Percolation theory can be generally applied to calculate the effective conductivity for a mixture of conductive components having a certain conductivity of R and permittivity of R → , and dielectric components with a finite permittivity, C , and conductivity, C = 0, corresponding to the resistive and capacitive sites/bonds, respectively. ...
... The impedance can be obtained by solving the complex linear equations established by applying the Kirchhoff's current law to each particle, denoting a node of the network. 17 With this numerical strategy, we calculated the AC responses of RC network with various geometries. Figure 2 shows the dependence of normalised impedance module,˜ Z on normalised frequency,˜ , based on 2D square, SC (simple cubic), random and FCC (face-cantered cubic) packing structures. ...
Article
Large-scale dynamical systems, no matter whether possessing interconnected appearances, are frequently modeled as networks. However, few of them have been studied from the perspective of the frequency domain, mostly due to the complexity of evaluating their frequency response. That gap is filled by this paper which proposes algorithms computing a general class of self-similar networks' frequency response and transfer functions, no matter they are finite or infinite, damaged or undamaged. In addition, this paper shows that for infinite self-similar networks, even when they are damaged, fractional-order and irrational dynamics naturally come into sight. Most importantly, this paper illustrates that for a network under different operating conditions, its frequency response would form a set of neighboring plants, which sets the basis of applying robust control methods to dynamic networks.
Article
Relationships between microstructure characteristics and effective transport properties of granular materials are crucial for many real-world applications. In the present paper microstructure-property relationships of sphere packings are investigated by means of modeling and simulation. Virtual microstructures are generated with the random close packing algorithm. This algorithm provides initial systems of randomly distributed, non-overlapping and densely-packed spheres of a given class of polydisperse size distributions. Next, the initial sphere packing is further densified until a certain criterion is reached, namely a predefined mean contact angle. In this way, we obtain a large database of slightly overlapping sphere systems. Subsequently, effective transport properties of the sphere systems (solid) and their complementary sets (pores) are determined using the computationally efficient resistor network method. Finally, the generated virtual microstructures are used to establish formulas expressing effective transport properties of the considered sphere packings in terms of the mean contact angle and the standard deviation of the particle radii.
Article
This paper introduces mechanical networks as a tool for modeling complex unidirectional vibrations. Networks of this type have branches containing massless linear springs and dampers, with masses at the nodes. Tree and ladder configurations are examples demonstrating that the overall dynamics of infinite systems can be represented using implicitly defined integro-differential operators. Results from the proposed models compare well to numerical results from finite systems, so this approach may have advantages over high-order differential equations.
Article
Full-text available
The universality of the ac electrical transport in granular metals has been scarcely studied and the actual mechanisms involved in the scaling laws are not well understood. Previous works have reported on the scaling of capacitance and dielectric loss at different temperatures in Co-ZrO2 granular metals. However, the characteristic frequency used to scale the conductivity spectra has not been discussed, yet. This report provides unambiguous evidence of the universal relaxation behavior of Pd-ZrO2 granular thin films over wide frequency (11Hz–2MHz) and temperature ranges (40–180K) by means of Impedance Spectroscopy. The frequency dependence of the imaginary parts of both the impedance Z″ and electrical modulus M″ exhibit respective peaks at frequencies ωmax that follow a thermal activation law, ω_max∝exp(T^1/2). Moreover, the real part of electrical conductivity σ′ follows the Jonscher’s universal power law, while the onset of the conductivity dispersion also corresponds to ωmax. Interestingly enough, ωmax can be used as the scaling parameter for Z″, M″ and σ′, such that the corresponding spectra collapse onto single master curves. All in all, these facts show that the TimeTemperature Superposition Principle holds for the ac conductance of granular metals, in which both electron tunneling and capacitive paths among particles compete, exhibiting a well-characterized universal behavior
Article
Full-text available
This paper introduces a concept that allows the creation of low-resistance composites using a network of compliant conductive aggregate units, connected through contact, embedded within the composite. Due to the straight-forward fabrication method of the aggregate, conductive composites can be created in nearly arbitrary shapes and sizes, with a lower bound near the length scale of the conductive cell used in the aggregate. The described instantiation involves aggregate cells that are approximately spherical copper coils-of-coils within a polymeric matrix, but the concept can be implemented with a wide range of conductor elements, cell geometries, and matrix materials due to its lack of reliance on specific material chemistries. The aggregate cell network provides a conductive pathway that can have orders of magnitude lower resistance than that of the matrix material - from 10(12) ohm-cm (approx.) for pure silicone rubber to as low as 1 ohm-cm for the silicone/copper composite at room temperature for the presented example. After describing the basic concept and key factors involved in its success, three methods of implementing the aggregate into a matrix are then addressed - unjammed packing, jammed packing, and pre-stressed jammed packing - with an analysis of the tradeoffs between increased stiffness and improved resistivity.
Article
We investigate polarization-resolved terahertz (THz) transmission through a doped polythiophene film consisting of partially oriented polymer chains. The THz complex conductivities are found to be significantly larger for polarization parallel to the principal direction of orientation than for polarization perpendicular to it, but involve no change in spectral shape with polarization. This indicates that charge transport occurs mainly along polythiophene chains with their in-plane angle distribution, ruling out a possible interchain contribution, whose spectral shape should be sensitive to polarization.
Article
This study investigates the effects of sintering in random composite packings of spherical particles, with focus on structures with relatively low densification where the sintering is used to obtain desired catalytic and transport properties. The effects of the degree of sintering (particle–particle contact angle) and of additional porosity created by pore-former particles on coordination numbers and percolation probabilities are addressed by using both the extended percolation theory and the drop-and-roll numerical reconstruction method. The comparison of the two methods allows the assessment of two key assumptions on which percolation theory relies when applied to sintered structures: i) particles retain their position during the sintering, ii) coordination numbers in the sintered structure are evaluated as the particles were rigid spheres. The former assumption is assessed by calculating, using the drop-and-roll method, the fraction of collapsing particles, which belong to clusters completely surrounded by pore-formers. Numerical simulations show that the fraction of collapsing particles is less than 1% for porosities as high as 60%. Within the range of validity of the first assumption, the theoretical and numerical methods predict that pore-formers decrease the number of contacts and so the percolation probability. The latter assumption is assessed by comparing simulated and theoretical results for different contact angles. The comparison shows that for contact angles smaller than 15° particles behave as they were rigid. For contact angles larger than 23°, sintering effects are no longer negligible, leading to a relative error between the two methods larger than 10% in contact number estimations. This study also shows that percolation theory and numerical simulations provide very similar results over a wide range of conditions, suggesting that the two methods can be used interchangeably for describing sintered random packings.
Article
Optimization of the microstructure of porous electrodes plays an important role in the enhancement of the performance of solid oxide fuel cells. For this, microstructural models based on percolation theory have proven useful for the estimation of the effective material properties of the electrode material, assumed to consist of a binary mixture of spherical electron and ion conducting particles. In this work, we propose an extension of prior approaches for calculating the effective size of the three-phase boundary, which we judge to be physically more sound and, in particular, well suited for characterizing mixtures of particles of different sizes. This approach is then employed in a one-dimensional cell level model encompassing the entire set of processes of gas transport, electronic and ionic conduction as well as the electrochemical reactions. The impact of the electron and ion conducting particle sizes, their volume fraction and their size ratio on the performance of the fuel cell are investigated in a parametric study. Under certain conditions, cathode microstructures having electronic conducting particles of size different from that of the ionic conducting particles become preferable and yield a higher maximum power density when compared to the best possible configuration of monodisperse particles.
Article
This article presents a review of the current understanding of the mechanics of random fiber networks. The discussion refers to athermal fiber networks, for which the governing functional is the system enthalpy, as well as to molecular networks, in which thermal fluctuations are important. Fiber networks are broadly encountered in everyday life as paper, insulation and damping materials, and as the essential component of some consumer products, while molecular networks form the structure of biological and non-biological materials such as the cytoskeleton, connective tissue, gels and rubber. The mechanics of these materials is defined by the structure of the network and by the mechanical behavior of individual filaments. The structure is characterized by a number of parameters, such as the density, filament orientation, fiber waviness, density and nature of cross-links, which are discussed in the first part of the article. The constitutive behavior of individual filaments is defined by their bending and axial stiffness, and by the response to thermal fluctuations. The constitutive response of the cross-links plays a central role in defining the system-scale mechanical behavior. The analysis of the network mechanics is divided into two parts addressing cross-linked and entangled (non-cross-linked) networks. Both molecular and athermal cross-linked networks are discussed, while only the literature on athermal entangled networks is included. Flexible and semi-flexible fibers are considered, with special attention given to the second category. The constitutive behavior under small and large deformations, including attempts to define continuum representations of the discrete system, is reviewed. A number of open issues are discussed in closure.
Article
A new method for the prediction of coordination numbers in random packings of rigid spherical particles is presented, consisting of improvements of basic relationships of percolation theory for the determination of numbers of contacts, percolation thresholds and probability of connection in binary mixtures and their extension to multicomponent and polydisperse mixtures. The proposed model is critically compared with previous percolation theories, showing a satisfactory agreement with experimental data and computer simulations of random packings over a wide range of particle sizes and compositions for both binary and multicomponent/polydisperse mixtures.
Article
This study extends the use of time domain reflectometry (TDR) in geotechnical engineering, a technique originally developed to locate faults in transmission lines. Different elements of the TDR technique are developed, including design of TDR probes, probe installation/test methodology, and relationships between TDR measured dielectric constant and water content of soil. A coaxial probe is developed that is used for measuring the dielectric constant of soil prepared in a cylindrical cell or compaction mold. A multiple-rod field probe is developed that modifies previously developed multiple-rod probes and extends their capability for measuring the in-place dielectric constant of soil. An analytical solution is developed to determine the sampling volume and spatial bias of the TDR measurement. The solution is extended to study the effect of soil disturbance and presence of air gaps due to probe insertion. Experimental results validate the solutions. New relationships are proposed between dielectric constant and water content to eliminate some of the limitations of the existing calibration relationships. Several possible applications of the developed probes, test methodology, and calibration equations for measuring water content and density of soil are illustrated.
Article
We review theoretical and experimental studies of the AC dielectric response of inhomogeneous materials, modelled as bond percolation networks, with a binary (conductor-dielectric) distribution of bond conductances. We first summarize the key results of percolation theory, concerning mostly geometrical and static (DC) transport properties, with emphasis on the scaling properties of the critical region around the percolation threshold. The frequency-dependent (AC) response of a general binary model is then studied by means of various approaches, including the effective-medium approximation, a scaling theory of the critical region, numerical computations using the transfer-matrix algorithm, and several exactly solvable deterministic fractal models. Transient regimes, related to singularities in the complex-frequency plane, are also investigated. Theoretical predictions are made more explicit in two specific cases, namely R-C and RL-C networks, and compared with a broad variety of experimental results, concerning, for example, granular composites, thin films, powders, microemulsions, cermets, porous ceramics and the viscoelastic properties of gels.