Electrical transport in networked materials occurs through percolative clusters composed of a random distribution of two kinds of interconnected elements: elementary nanostructures and nanostructure-nanostructure junctions. Rationalizing the contribution of these microscopic elements to the macroscopic resistance of the system is a fundamental issue to develop this class of materials and related devices. Focusing on networks composed of high-aspect-ratio nanostructures, such as nanowires (NWs) or nanotubes (NTs), these concepts are still raising controversy in modeling and interpretation of experimental data. Despite these incongruences and the large variations induced by disorder in the electrical properties of such networked systems, this work shows that the ratio between the junction and the nanostructure resistance is nearly the same at the microscopic and macroscopic levels, regardless of the network features. In other words, this means that we may assess the relative contribution of nanostructures and junctions to the macroscopic network resistance directly from the knowledge of its microscopic building blocks. Based on experimental data available in the literature, this result is proven to hold for different materials and network densities, ranging from inorganic NWs to organic carbon NTs and from the percolation critical density nc up to, at least, five times nc, respectively.

Nanostructures with (quasi) one-dimensional morphology, such as nanowires (NWs) or nanotubes (NTs), have been thoroughly investigated in recent years due to their unique structural and functional properties. In particular, random networks of these nanomaterials revealed to be particularly successful in achieving excellent functionalities using cheap and easy fabrication methodologies.1,2 For example, the efficient charge transport intrinsic in these nanostructures allowed us to realize very sparse matrixes featuring low macroscopic resistance, suitable for outstanding transparent electrodes.3 Networks with a high porosity and surface area revealed to be effective as dye-sorbent layers for dye-sensitized solar cells,4 electrodes for supercapacitors,5 and sensing elements for gas-6 and bio-sensors.7 Assemblies of Ag NWs with tuned NW-NW junction properties further revealed promising memristor functionalities.8 

Any progress in the development of these materials and devices strictly depends on our capability to unravel the relationship between their macroscopic and microscopic properties.

This means understanding the intricate interplay between the spatial distribution of the network building blocks and their electrical and morphological characteristics, which determine the macroscopic electrical features of the network.

Different techniques have been developed to shed light on this complex scenario. These include spatially resolved electroluminescence measurements, suitable to visualize the active conduction paths,9 TEM-tomography techniques, able to provide a 3D reconstruction of the percolative structure of real devices,10 and procedures to encode SEM images of 2D nanowire networks and combine the extrapolated matrixes with Kirchhoff-law algorithms solving the network resistance.11 

Besides the remarkable results achieved using these techniques, a major question remains whether the knowledge on random networks should be separately determined network by network or if we may find any general rule, though approximate or partial, relating the macroscopic network resistance to the resistances of its elementary building blocks.

From an electrical point of view, the network is composed of a disordered distribution of two kinds of microscopic elements: the nanostructures with wire/tube morphology and the nanostructure-nanostructure junctions.

In the literature, it is often assumed that the macroscopic resistance of the network, Rnet, is mainly determined by junction elements, considering negligible the resistance of elementary NWs/NTs.12–14 This approximation finds its basis on resistivity arguments. Indeed, at the level of single microscopic elements, it has been proven for a variety of systems that the junction resistivity (ρj) is much larger than the intrinsic resistivity of the nanostructures (ρn): ρj ≫ ρn.15–17 Besides this, a direct extension of such an evidence to the respective microscopic resistances, Rj and Rn, should be carefully evaluated. Several recent studies report about networks featuring RjRn or even Rj < Rn. For example, this is the case of highly efficient transparent electrodes, in which the junction resistance has been minimized through thermal or radiation-exposure treatments.3,17 A similar situation is encountered working with long NWs/NTs, whose length, according to 2nd Ohm's law, may be large enough to counterbalance the resistivity effects.16,18,19

To contribute answering these still open-questions, the present work will show that, besides the large variation observed in the macroscopic resistance of random networks and discrepancies still existing among theoretical models, experimental results reported in the literature fairly converge to a common, simple law, which resumes network effects through a single parameter, αnet,

(1)
(1a)
It is useful to introduce Rnet-j and Rnet-n, which represent the macroscopic junction and nanostructure components of Rnet, such that Rnet can be expressed as the series of these two terms (Rnet = Rnet-j + Rnet-n). As it will be shown later, Eq. (1a) may be expressed in the following equivalent form:

(1b)

Equations (1) mean that the network effects apply almost equally to all the network building blocks (nanostructures and junctions). This is to say that the relative contribution of junctions and nanostructures to the macroscopic network resistance, which may be quantified through the junction-to-nanostructure resistance-ratio, is almost the same at the macroscopic (Rnet-j/Rnet-n) and microscopic level (Rj/Rn).

To prove Eqs. (1), it is worth starting from the simplest electrical circuit, composed of a flat substrate with two electrodes and NWs/NTs randomly dispersed therein. As the density of the nanostructures increases up to and beyond the percolation threshold nc, a cluster builds up providing the electrical connection between the electrodes. The macroscopic electrical properties of the so-formed device are determined by converting such a disordered distribution of interconnected microscopic nanostructures and junctions into a network of junction-type and nanowire-type (nanotube-type) resistors, as schematically shown in Fig. 1.

FIG. 1.

Schematic representation of a random network composed of NWs or NTs (a) and the equivalent electrical circuit of the cyan-colored network portion (b). Contact points between different NWs/NTs divide these nanostructures into fragments. Each fragment features a wire/tube-type resistance that, according to Ohm's law, is a fraction of the elementary NW/NT resistance, Rn. Nanostructure-nanostructure contacts are represented by their equivalent junction resistance, Rj.

FIG. 1.

Schematic representation of a random network composed of NWs or NTs (a) and the equivalent electrical circuit of the cyan-colored network portion (b). Contact points between different NWs/NTs divide these nanostructures into fragments. Each fragment features a wire/tube-type resistance that, according to Ohm's law, is a fraction of the elementary NW/NT resistance, Rn. Nanostructure-nanostructure contacts are represented by their equivalent junction resistance, Rj.

Close modal

For any device/network, under the hypothesis of a nearly homogeneous distribution for the microscopic properties of elementary network components, Rj and Rn are well represented by average values, which can be determined through local measurements on elementary components.20,21 Similarly, Rnet-j and Rnet-n will be expressed through their own numerical values. So far, whatever the networking degree, network effects can be conveniently expressed by means of two proportionality factors αnet-j and αnet-n,

(2)
(2a)
(2b)
From this point of view, proving Eqs. (1) means looking for any underlying relationship between αnet-j and αnet-n (in particular, αnet-j ≈ αnet-n).

To this aim, an interesting set of data is provided by Gomes da Rocha and co-workers,11 who measured the macroscopic resistance (Rnet) of several networks composed of Ag NWs and fitted these data with the equation Rnet = αnet-j Rj + Rnet-n.

To look at these results in terms of Eqs. (1) and (2), a further analysis is here carried out looking for any underlying relationship between Rnet-n and αnet-j. The values of Rnet-n and αnet-j characterizing each network are retrieved from Ref. 11, Table A1 of the Electronic Supplementary Information (there named R0 and a0, respectively).

As shown in Fig. 2(a), a linear dependence emerges, Rnet-n = k αnet-j, where k is the slope of the linear fit. Fitting with a second order polynomial returns a coefficient for the squared term, that is, lower than its uncertainty, thus suggesting the lack of evidence for any relationship more complex than linear. Considering Eqs. (2), the slope can be expressed as k = Rn αnet-n/αnet-j. Given the average resistance of the microscopic NWs, Rn ≈ 77 Ω, as calculated from their average length (Lnw ≈ 6.7 μm), radius (25 nm), and resistivity (22.6 nΩm),11 the slope k 32 Ω means αnet-j/αnet-n ≈ 2.38. However, the network structure divides each NW into fragments, and thus, 6.7 μm represents the upper limit for the average effective length traveled by electrons in the NW bodies (leading to the upper bound αnet-j ≤ 2.38 αnet-n). Considering the average distance between neighboring NWs (≈0.37 Lnw, Ref. 22) as a more reliable reference value for such an effective travelling-length, the value of k leads to a NW resistance Rn ≈ 29 Ω and an almost unitary αnet-j/αnet-n ratio. This means that network effects can be summarized through a single parameter αnet (≈αnet-nαnet-j) as stated in Eqs. (1).

FIG. 2.

Analysis of the Rnet-n vs αnet-j (a) and the 1/Rnet vs n-nc (b) relationships for Ag NW networks. Experimental data (full-circles) are from Ref. 11. Dashed lines represent the linear fits.

FIG. 2.

Analysis of the Rnet-n vs αnet-j (a) and the 1/Rnet vs n-nc (b) relationships for Ag NW networks. Experimental data (full-circles) are from Ref. 11. Dashed lines represent the linear fits.

Close modal

It is further worth noting that a single value for k (and thus the relationship αnet-nαnet-j) works for the whole set of networks of Ref. 11. In terms of nondimensional density n, defined as n=NLnw2, where N is the number of NWs per unit area and Lnw is as above, it means that Eqs. (1) hold for n ranging from 5.4 to 28.7, i.e., from about the critical density of 2D percolating systems (nc ≈ 5.637),23 up to about 5nc.

This is a nontrivial result. Indeed, the modeling and explanation of networked materials still present ambiguities. For example, this is the case of the 1/Rnet vs n relationship, which is the fundamental argument adopted to identify the transport regime of networked materials. It is well established that electrical transport undergoes a gradual transition from the percolating regime, holding for n close to nc, to the continuum one, observed for large densities. The former is characterized by the power law of percolation, 1/Rnet ∝ (n-nc)p, where p is the percolation exponent, and by a scattered voltage distribution along the network. The latter features a linear 1/Rnet vs n dependence and a regular, almost linear voltage distribution.24–29 Besides such a general agreement, divergences emerge if we try to interpret the physical meaning of these laws or if we attempt to track quantitatively the transition between the two regimes. For example, some theories and experiments report the typical percolation features occurring also for n >20nc,24,25,30,31 while others predict/observe the continuum regime already for n >2nc.26–28 Analyzing the networks of Fig. 2(a) in terms of 1/Rnet vs n (data are available from Ref. 11) leads to the results shown in Fig. 2(b). The plot is very scattered, and its fitting is not exempt from ambiguity. As detailed in the supplementary materials, fitting with different models, spanning from those based on percolation theory24,25 till those based on physical and geometrical arguments,26,32,33 returns similar correlation coefficients.

This situation arises from a combination of theoretical and experimental issues. From a conceptual point of view, the intrinsic complexity of the argument forces to adopt approximations that inevitably limit the prediction capability. On the other hand, experimental issues such as the finite size of real devices or inhomogeneity affecting the properties of elementary NWs/NTs, such as their diameter, length, orientation, or curviness, strongly complicate the modeling and simulation of real systems.30,34–36

These effects are expected to apply to Ag NW networks studied in Fig. 2 as well, but, as shown by representative volume element (RVE) arguments, the large scattering observed in Fig. 2(b) is likely to originate also from the intrinsic disordered of such a set of networks. Introducing Ldev as the spacing between the electrodes of the device, RVE is the minimum value of the nondimensional length Ldev/Lnw beyond which the network works as a homogeneous material for a given property.37 As detailed in the supplementary material, referring to Rnet as the target property, all the networks analyzed in Fig. 2 feature an RVE larger than the experimental Ldev/Lnw, suggesting these systems being highly inhomogeneous, with properties varying from network to network.

The mentioned challenges and divergences have been discussed in terms of 1/Rnet vs n, to which typically refer literature models, but these may be reasonably considered the ground of theoretical disagreements concerning the junction-to-nanostructure resistance-ratio.

While most of the models focus on Rnet, only a few provide an expression for Rnet-j and Rnet-n, which allows discussing the relationship between αnet-j and αnet-n. Among these, the geometrical model developed by Kumar et al., which holds for n 2nc,32 predicts αnet-j/αnet-n increasing with n as a consequence of the incremental fragmentation of NWs/NTs

(3)

As shown in Fig. 3, according to Eq. (3), αnet-j/αnet-n is always appreciably larger than unity, reaching a value as large as 20 for n 30 (the limit of the experimental data reported in Fig. 2). On the other hand, the generalized percolation model developed by Žeželj and Stanković,24 which works for the whole range of n values shown in Fig. 3, predicts an opposite behavior, with αnet-j/αnet-n decreasing with increasing n,

(4)

Since p varies from 1 to 2 depending on the Rj/Rn ratio and n,24 predictions of the Žeželj and Stanković model are reported in Fig. 3 for the universal electrical exponent (p =1.29) as well as for its two extremes. Besides the p dependence of the curve αnet-j/αnet-n vs n, all the curves based on the latter model show a common behavior, opposite to the geometrical model. Increasing n from nc to 5nc, αnet-j/αnet-n decreases from 2 to 0.5 (for p =1.29).

FIG. 3.

NW/NT density (n) dependence of the αnet-jnet-n ratio calculated according to different theories: the geometrical model, Eq. (3),32 and the model proposed by Žeželj and Stanković, Eq. (4),24 calculated for the percolation exponent equal to the universal value, p = 1.29, as well as for its extreme values, p = 1 and p = 2.

FIG. 3.

NW/NT density (n) dependence of the αnet-jnet-n ratio calculated according to different theories: the geometrical model, Eq. (3),32 and the model proposed by Žeželj and Stanković, Eq. (4),24 calculated for the percolation exponent equal to the universal value, p = 1.29, as well as for its extreme values, p = 1 and p = 2.

Close modal

Despite these divergences, both theories predict αnet-j/αnet-n which is unaffected by the finite-size of the device. In percolation theory, finite-size effects have a variety of consequences, which include smoothing the insulating-conducting transition at nnc and the modulation of the nc value.36,38,39 Interestingly, the percolation model24 predicts the same finite-size correction for Rnet-j and Rnet-n and it does not affect the αnet-j/αnet-n ratio in Eq. (4). The geometrical model32 does not present any finite-size term, which is also absent in Eq. (3).

Equations (1) are also consistent with experiments carried out with carbon nanotubes.

For example, impedance spectroscopy has been applied to decouple Rnet-j and Rnet-n in polymeric photonic crystals intercalated with single walled carbon nanotubes (SWCNTs). The electrical and geometrical properties of these materials were tuned through the modulation of SWCNT loading and postsynthesis annealing treatments.40 In particular, the duration of the thermal treatment was suggested to modify the electrical properties of CNTs and CNT-CNT junctions directly at the microscopic level (through CNT relaxation arising from viscosity-modifications occurring in the embedding polymer).40 Following this suggestion, samples prepared by means of different SWCNT loadings but the same thermal treatment can be reasonably considered to be composed of different networks of similar elementary nanostructures and junctions. In this view, the large variations (more than one order of magnitude) measured for both Rnet-j and Rnet-n as the SWCNT loading increases by a factor 2 [Figs. 4(a) and 4(b)] can be ascribed to networking effects, i.e., to variations in αnet-n and αnet-j.

FIG. 4.

Analysis of SWCNT networks embedded in polymeric photonics crystals. Annealing-time and SWCNT loading dependence of the macroscopic resistances Rnet-j (a), Rnet-n (b), and their ratio Rnet-j/Rnet-n (c). Experimental data are from Ref. 40. (a) and (b) are adapted with permission from Imai et al., Soft Matter 8, 6280 (2012). Copyright 2012 The Royal Society of Chemistry.

FIG. 4.

Analysis of SWCNT networks embedded in polymeric photonics crystals. Annealing-time and SWCNT loading dependence of the macroscopic resistances Rnet-j (a), Rnet-n (b), and their ratio Rnet-j/Rnet-n (c). Experimental data are from Ref. 40. (a) and (b) are adapted with permission from Imai et al., Soft Matter 8, 6280 (2012). Copyright 2012 The Royal Society of Chemistry.

Close modal

Despite such a large variation measured for both macroscopic resistances, their ratio Rnet-j/Rnet-n shows a much narrower distribution. For example, for the shorter thermal treatment (1 h), Rnet-j/Rnet-n ranges between 40 and 50. A longer annealing slightly broadens the range of values spanned by Rnet-j/Rnet-n, but the spreading of these values remains much smaller than the variations observed for Rnet-j and Rnet-n [Fig. 4(c)].

A similar situation has also been observed with random assemblies of SWCNTs. Impedance spectroscopy has been used to compare networks realized with SWCNTs at different densities, corresponding to about 1.6 and 6.4 times nc. These measurements showed large variations for both Rnet-j (≈24 kΩ and ≈1 kΩ) and Rnet-n (≈6.6 kΩ and ≈0.5 kΩ), while the ratio Rnet-j/Rnet-n was nearly the same, about 3 and 3.5.41 

Looking at the results of these two CNT-papers as a whole, if we assume αnet-j and αnet-n as two uncorrelated parameters, the Rnet-j/Rnet-n ratio would be reasonably expected to feature variations comparable with those of Rnet-j and Rnet-n. Instead, its much more pronounced regularity, observed for very different systems, suggests the existence of a strong correlation between αnet-j and αnet-n. The lack of information such as the length of CNTs or the size of devices used in Refs. 40 and 41, joined with the wide variations reported in the literature for CNTs' Rj and Rn,35,41 makes it difficult to go beyond this level with the analysis of CNT data. Indeed, the method developed in Ref. 11 and applied to Ag NWs is almost unique in its capability to unravel the relationship between the microscopic and macroscopic network parameters. Even if data available for CNT systems are not sufficiently detailed to firmly conclude that the correlation between CNTs' αnet-j and αnet-n is the same as that found for Ag NW networks, i.e., αnet-n ≈ αnet-j, their results do not exclude, at least, this conclusion.

Further considering that Eqs. (1) revealed robust against inhomogeneity and large variations observed from network to network in Fig. 2; the same Eqs. (1) may be reasonably considered to hold in general for networks of (quasi) one-dimensional nanostructures (at least for densities up to about 5nc).

In summary, the present work shows that, despite the large fluctuations observed in the macroscopic resistance of random networks composed of high-aspect-ratio nanostructures and discrepancies still existing among theoretical models, experimental data acquired with different materials, including inorganic metallic NWs and organic (semi)conducting CNTs, converge to a simple, common law, expressed in Eqs. (1). Given the complexity of network effects and considering the sophisticated computational and/or experimental tools required to unravel these concepts,9–11 which are not usually readily available, the findings of the present work represent a simple and suitable tool for the broad community dealing with networked materials. In particular, Eq. (1b) allows expressing the relative macroscopic contribution of junctions and NWs/NTs directly from their microscopic values Rj and Rn. This is expected to be useful in the design and understanding of (semi)conducting networks. The validity of Eqs. (1) is here proved for network densities up to about 5nc. More in general, considering the good convergence obtained with experimental data characterized by a high degree of disorder and inhomogeneity, it may be foreseen to have Eqs. (1) working also for larger densities.

See the supplementary material for the full analysis of the Ag NW data in terms of 1/Rnet vs n and for the calculation of the representative volume element (RVE).

This work was partially funded by Lombardia Region and Fondazione Cariplo through the project EMPATIA@LECCO and by Lombardia Region and CNR through the project FHfFC.

1.
S. V. N. T.
Kuchibhatla
,
A. S.
Karakoti
,
D.
Bera
, and
S.
Seal
,
Prog. Mater. Sci.
52
,
699
(
2007
).
2.
A. V.
Kyrylyuk
,
M. C.
Hermant
,
T.
Schilling
,
B.
Klumperman
,
C. E.
Koning
, and
P.
van der Schoot
,
Nat. Nanotechnol.
6
,
364
(
2011
).
3.
E. C.
Garnett
,
W.
Cai
,
J. J.
Cha
,
F.
Mahmood
,
S. T.
Connor
,
M. G.
Christoforo
,
Y.
Cui
,
M. D.
McGehee
, and
M. L.
Brongersma
,
Nat. Mater.
11
,
241
(
2012
).
4.
P. S.
Archana
,
A.
Gupta
,
M. M.
Yusoff
, and
R.
Jose
,
Appl. Phys. Lett.
105
,
153901
(
2014
).
5.
P. J.
King
,
T. M.
Higgins
,
S.
De
,
N.
Nicoloso
, and
J. N.
Coleman
,
ACS Nano
6
,
1732
(
2012
).
6.
A.
Ponzoni
,
E.
Comini
,
G.
Sberveglieri
,
J.
Zhou
,
S. Z.
Deng
,
N. S.
Xu
,
Y.
Ding
, and
Z. L.
Wang
,
Appl. Phys. Lett.
88
,
203101
(
2006
).
7.
W. C.
Lee
,
H.
Lee
,
J.
Lim
, and
Y. J.
Park
,
Appl. Phys. Lett.
109
,
223701
(
2016
).
8.
H.
Du
,
T.
Wan
,
B.
Qu
,
F.
Cao
,
Q.
Lin
,
N.
Chen
,
X.
Lin
, and
D.
Chu
,
ACS Appl. Mater. Interfaces
9
,
20762
(
2017
).
9.
R.
Rother
,
S. P.
Schießl
,
Y.
Zakharko
,
F.
Gannott
, and
J.
Zaumseil
,
ACS Appl. Mater. Interfaces
8
,
5571
(
2016
).
10.
S. D.
Oosterhout
,
M. M.
Wienk
,
S. S.
van Bavel
,
R.
Thiedmann
,
L. J. A.
Koster
,
J.
Gilot
,
J.
Loos
,
V.
Schmidt
, and
R. A. J.
Janssen
,
Nat. Mater.
8
,
818
(
2009
).
11.
C.
Gomes da Rocha
,
H. G.
Manning
,
C.
O'Callaghan
,
C.
Ritter
,
A. T.
Bellew
,
J. J.
Boland
, and
M. S.
Ferreira
,
Nanoscale
7
,
13011
(
2015
).
12.
R. M.
Mutiso
and
K. I.
Winey
,
Phys. Rev. E
88
,
032134
(
2013
).
13.
A.
Ponzoni
,
V.
Russo
,
A.
Bailini
,
C. S.
Casari
,
M.
Ferroni
,
A.
Li Bassi
,
A.
Migliori
,
V.
Morandi
,
L.
Ortolani
,
G.
Sberveglieri
, and
C. E.
Bottani
,
Sens. Actuators, B
153
,
340
(
2011
).
14.
P. N.
Nirmalraj
,
P. E.
Lyons
,
S.
De
,
J. N.
Coleman
, and
J. J.
Boland
,
Nano Lett.
9
,
3890
(
2009
).
15.
J. W.
Orton
and
M. J.
Powell
,
Rep. Prog. Phys.
43
,
1263
(
1980
).
16.
D.
Hecht
,
L.
Hu
, and
G.
Grüner
,
Appl. Phys. Lett.
89
,
133112
(
2006
).
17.
D. P.
Langley
,
M.
Lagrange
,
G.
Giusti
,
C.
Jiménez
,
Y.
Breéchet
,
N. D.
Nguyen
, and
D.
Bellet
,
Nanoscale
6
,
13535
(
2014
).
18.
C.
O'Callaghan
,
C.
Gomes da Rocha
,
H. G.
Manning
,
J. J.
Boland
, and
M. S.
Ferreira
,
Phys. Chem. Chem. Phys.
18
,
27564
(
2016
).
19.
A.
Ponzoni
,
C.
Baratto
,
S.
Bianchi
,
E.
Comini
,
M.
Ferroni
,
M.
Pardo
,
M.
Vezzoli
,
A.
Vomiero
,
G.
Faglia
, and
G.
Sberveglieri
,
IEEE Sens. J.
8
,
735
(
2008
).
20.
A. T.
Bellew
,
H. G.
Manning
,
C.
Gomes da Rocha
,
M. S.
Ferreira
, and
J. J.
Boland
,
ACS Nano
9
,
11422
(
2015
).
21.
F.
Selzer
,
C.
Floresca
,
D.
Kneppe
,
L.
Bormann
,
C.
Sachse
,
N.
Weiß
,
A.
Eychmüller
,
A.
Amassian
,
L.
Müller-Meskamp
, and
K.
Leo
,
Appl. Phys. Lett.
108
,
163302
(
2016
).
22.
J.
Heitz
,
Y.
Leroy
,
L.
Hébrard
, and
C.
Lallement
,
Nanotechnology
22
,
345703
(
2011
).
23.
J.
Li
and
S.-L.
Zhang
,
Phys. Rev. E
80
,
040104(R)
(
2009
).
24.
M.
Žeželj
and
I.
Stanković
,
Phys. Rev. B
86
,
134202
(
2012
).
25.
26.
C.
Forró
,
L.
Demkó
,
S.
Weydert
,
J.
Vörös
, and
K.
Tybrandt
,
ACS Nano
12
,
11080
(
2018
).
27.
T.
Sannicolo
,
N.
Charvin
,
L.
Flandin
,
S.
Kraus
,
D. T.
Papanastasiou
,
C.
Celle
,
J.-P.
Simonato
,
D.
Muñoz-Rojas
,
C.
Jiménez
, and
D.
Bellet
,
ACS Nano
12
,
4648
(
2018
).
28.
E. S.
Snow
,
J. P.
Novak
,
P. M.
Campbell
, and
D.
Park
,
Appl. Phys. Lett.
82
,
2145
(
2003
).
29.
S.
He
,
X.
Xu
,
X.
Qiu
,
Y.
He
, and
C.
Zhou
,
J. Appl. Phys.
124
,
054302
(
2018
).
30.
J.
Hicks
,
J.
Li
,
C.
Ying
, and
A.
Ural
,
J. Appl. Phys.
123
,
204309
(
2018
).
31.
M. B.
Bryning
,
M. F.
Islam
,
J. M.
Kikkawa
, and
A. G.
Yodh
,
Adv. Mater.
17
,
1186
(
2005
).
32.
A.
Kumar
,
N. S.
Vidhyadhiraja
, and
G. U.
Kulkarni
,
J. Appl. Phys.
122
,
045101
(
2017
).
33.
C. A.
Ainsworth
,
B.
Derby
, and
W. W.
Sampson
,
Adv. Theory Simul.
1
,
1700011
(
2018
).
34.
D. P.
Langley
,
M.
Lagrange
,
N. D.
Nguyen
, and
D.
Bellet
,
Nanoscale Horiz.
3
,
545
(
2018
).
35.
L.-P.
Simoneau
,
J.
Villeneuve
,
C. M.
Aguirre
,
R.
Martel
,
P.
Desjardins
, and
A.
Rochefort
,
J. Appl. Phys.
114
,
114312
(
2013
).
36.
X.
Ni
,
C.
Hui
,
N.
Su
,
W.
Jiang
, and
F.
Liu
,
Nanotechnology
29
,
075401
(
2018
).
37.
F.
Han
,
T.
Maloth
,
G.
Lubineau
,
R.
Yaldiz
, and
A.
Tevtia
,
Sci. Rep.
8
,
17494
(
2018
).
38.
M.
Žeželj
,
I.
Stanković
, and
A.
Belić
,
Phys. Rev. E
85
,
021101
(
2012
).
39.
D.
Kim
and
J.
Nam
,
J. Appl. Phys.
124
,
215104
(
2018
).
40.
Y.
Imai
,
C. E.
Finlayson
,
P.
Goldberg-Oppenheimer
,
Q.
Zhao
,
P.
Spahn
,
D. R. E.
Snoswell
,
A. I.
Haines
,
G. P.
Hellmann
, and
J. J.
Baumberg
,
Soft Matter
8
,
6280
(
2012
).
41.
M. P.
Garrett
,
I. N.
Ivanov
,
R. A.
Gerhardt
,
A. A.
Puretzky
, and
D. B.
Geohegan
,
Appl. Phys. Lett.
97
,
163105
(
2010
).

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