Daniel P.
Langley‡
^{abc},
Mélanie
Lagrange‡
^{a},
Ngoc Duy
Nguyen
^{b} and
Daniel
Bellet
*^{a}
^{a}Univ. Grenoble Alpes, CNRS, Grenoble INP, LMGP, 38000 Grenoble, France. E-mail: daniel.bellet@grenoble-inp.fr
^{b}Département de Physique, CESAM/Q-MAT, SPIN, Université de Liège, B-4000 Liège, Belgium
^{c}Swinburne University of Technology, Department of Telecommunications, Electrical, Robotics and Biomedical Engineering, John Street, 3122 Hawthorn, Victoria, Australia

Received
18th March 2018
, Accepted 12th June 2018

First published on 19th June 2018

Planar networks composed of 1-dimensional nanometer scale objects such as nanotubes or nanowires have been attracting growing interest in recent years. In this work we directly compare the percolation threshold of silver nanowire networks to predictions from Monte Carlo simulations, focusing particularly on understanding the impact of real world imperfections on the percolation onset in these systems. This work initially determines the percolation threshold as calculated from an ideal system using Monte Carlo methods. On this foundation we address the effects of perturbations in length, angular anisotropy and radius of curvature of the 1-dimensional objects, in line with those observed experimentally in purposely fabricated samples. This work explores why two-dimensional stick models in the literature currently underestimate the percolation onset in real systems and identifies which of the network's features play the most significant role in that deviation.

## Conceptual insightsThis work provides insight into the impact of imperfections on the percolation of 1D objects in a 2D network. This work combines simulations, theory and real world networks to explore the onset of percolation in 2D networks of silver nanowires. Three imperfections are recognised: length distributions, angular anisotropy and wire curvature. This work extends the previous literature to explain the discrepancy between theoretically predicted percolation onset and that observed experimentally. In light of this work it is clear that the angular distribution has the highest impact on network formation and that in a counter intuitive observation a length distribution actually works to lower the required nanowire density to form a percolating network. |

Although the technological interest is relatively recent, stick percolation in two-dimensional (2D) systems has already been the subject of research for many years. In their seminal paper of 1974, Pike and Seager^{13,14} addressed the general problem of percolation in a variety of systems. Their models involved the use of two main parameters which are the network density n, defined as the number of sticks per unit area, and the stick length L. In the process of building a denser network by increasing the number of wires with random orientation on a planar substrate, the percolation threshold is defined as the first occurrence of a set of intersecting sticks spanning the whole system and electrically connecting two parallel electrodes placed at opposite sides of the system. The network density for which the probability of finding a spanning cluster has the value of 0.5 is defined as the critical density n_{c}. Li and Zhang^{15} studied stick percolation by Monte Carlo simulations with much higher accuracy and showed, for system size much larger than L, that:

n_{c}L^{2} = 5.63726. | (1) |

σ ∝ (n − n_{c})^{γ} | (2) |

In the present work, we propose a more general approach, which incorporates the effects of three experimentally observed imperfections on the percolation onset. To achieve this, we develop a computational protocol based on Monte Carlo simulations and apply it to a network made of identical and randomly oriented sticks. In addition we explore the impact of imperfections that can be found in real nanowire networks on the percolation threshold. In real devices, imperfections can be observed as a non-uniform distribution of nanowire length, orientation and curvature. Our algorithms are based on the formalism proposed by Newman and Ziff^{20,21} which describes the formation of nanowire clusters and the identification of the conducting paths, renumbering the smaller clusters rather than the percolating cluster, thus enabling much faster modelling of the system than previous approaches in the literature. In this work comparing the theoretical results to the experimental electrical behaviour of real systems then validates this formalism. From there, a comparative study between simulations and experiments based on Ag nanowire networks has been carried out, allowing us to explore the mechanisms responsible for the electrical resistance of 2D metallic nanowire networks and how it relates to the onset of percolation which was analysed as function of network density, nanowire length distribution, non-isotropic angular distributions as well as nanowire curvature. Such a study of the percolation in perfect and imperfect nanowire networks based on computer simulation and experimental observations aims at improving the understanding of percolation mechanisms of 1D objects in general.

Fig. 1a shows a typical layout of an ideal network of nanowires as generated by our algorithms. By definition, percolation in this system occurs when a cluster of sticks gets connected to both the top and bottom edges of the system. From a large number of simulations (typically 10^{5} or 10^{6}) we can build up a statistical analysis of the network density at which the nanowires first span the system. This allows us to generate a percolation probability distribution function, as shown in Fig. 1b (black curve). In a subsequent step it is then possible to plot the cumulative probability distribution function (blue curve), from where the percolation threshold, defined as the network density at which the probability P of finding a network to be percolating is 0.5, can be extracted. This well-established definition is often used in the literature^{3,5,13–15} and is indicative of a transition from a non-percolating to a percolating system.

In order to validate our simulations we compared our results to those of Li and Zhang^{15} and we determined that for the ideal system where all nanowires have the same length with completely isotropic angular distribution, identical conclusions could be drawn from our simulations. From this data set we also extracted the values of N_{P} corresponding to the number of nanowires at percolation probabilities of P = 0.1, 0.25, 0.75 and 0.99 on a specified area. This additional data is displayed in Fig. 1c and shows a small deviation as expected from the theoretical value for P = 0.5.

One of our objectives is to investigate further these probability distribution functions by extracting useful information on the densities required to achieve a broader range of percolation probabilities. For example, in industrial applications it is more realistic to consider networks with stick densities leading to a probability of percolation as high as 0.99. In this case, such a high threshold is more viable as the likelihood of having a percolating system will be much higher and devices will be more reliable and robust, knowing that the requirements related to AgNW networks physical properties depend strongly on the application.^{9}

Each perturbation to the ideal system was then introduced in the form of a normal distribution for the corresponding variable. The width of these distributions is defined by a distribution parameter D_{L}, D_{A} and D_{RC} for the length, angular and radius of curvature distributions, respectively. These distribution parameters are used to generate normal distributions of length D_{L} and of radius of curvature D_{RC} as defined in eqn (S1)–(S3) (ESI†) where increasing the distribution parameter leads to an increase in the width of the distribution and results in a transition from the delta function of the ideal simulation towards an increasingly wide normal distribution. In contrast, an increase of the angular distribution parameter D_{A} results in a smooth transition from a uniform distribution of angles to a delta function (i.e. from the isotropic case where all wires are randomly oriented to the case where they would be all parallel to each other). As the values of the distribution parameters increase, the system deviates further from that of the ideal case.

The average wire length was kept constant. Yet, as D_{L} increases, the width of the normal distribution increases such that 4 standard deviations fall within D_{L}L of the average wire length (L) and the critical density n_{c} decreases as shown in Fig. 1d. As D_{A} increases the allowed angle range gets restricted, resulting in an increase in the anisotropy of the system and an increase in the percolation threshold as seen in Fig. 1e. The parameter D_{RC} is applied as a perturbation on the radius of curvature with three simple assumptions: (i) a 37.5 μm long nanowire with a radius of curvature of 10000 μm is a fair approximation of a straight nanowire, (ii) from direct observation of fabricated networks, wires are very unlikely to form a curve tighter than a semicircle, thus the minimum allowed radius was fixed to L/π, and (iii) very short nanowires (less than 10 μm) do not appear to bend, hence short wires were restricted to high radius of curvature. Therefore the perturbation is applied in such a way that the nanowires are initially assumed to be effectively straight with a radius of curvature of 10000 μm. Fig. 1f reports the influence of AgNW curvature on the critical density: a slight increase of n_{c} is observed when the curvature is increased although AgNW curvature does not appear as a prevailing parameter compared to the two others. A full definition of the distributions applied to the system can be found in the ESI† along with some representative images of the systems generated under such conditions (Fig. S1–S3, ESI†). For a fair comparison between real networks and the simulations reported in Fig. 1 our experimental work focused on the quantitative determination of the three distribution parameters mentioned above and on the experimental observation of the percolative nature of real networks by measuring their electrical resistance as function of network density for different AgNW sizes (average length and diameter).

Fig. 2 (a) Variation of the electrical resistance of AgNW-based networks of different densities measured in situ during a thermal annealing in air, operated from room temperature with a heating ramp rate of 2 °C min^{−1} (nanowire type: Ag117). (b) Minimum experimental resistance values associated to percolating networks made of Ag117, during thermal ramp as function of the network density. The fitting curve is calculated with eqn (3). (c) Silver nanowire networks density versus nanowire length diagram. The square red symbols are associated with non-percolating networks (i.e. as-measured non-conductive networks) while circle green markers are associated with conductive networks. The green markers sizes are proportional to the natural logarithmic values of the measured minimum electrical resistance R_{min}. The continuous line corresponds to the critical density n_{c} for uniform, perfect and infinite networks (eqn (1)). |

The experimental minimum resistance R_{min} measured during a thermal ramp can be obtained for a series of similar specimen with different network densities n. These values, reported in Fig. 2a, were measured in situ at elevated temperatures. For the proper comparison of network resistances it is better to compare resistance values at room temperature. To determine the minimum resistance at room temperature, one can extrapolate from the high temperature using the thermal coefficient as reported previously.^{24,26} The extrapolated values at room temperature are reported in Fig. 2b. From eqn (2) it can be shown that R_{min} depends on n and can be written as follows:

R_{min}(n) = R_{inst} + K·(n − n_{c})^{−γ} | (3) |

The data shown in Fig. 2a and b are associated to networks composed of the same nanowires type (Ag117). Experimental data were similarly extracted from networks made from the other nanowire types. These results are plotted in Fig. 2c, which represents the network densities of both percolating and non- percolating networks versus the average length (as determined by SEM image analysis) of the nanowires used. The full line illustrates eqn (1), which corresponds to the characteristic of a perfect theoretical network. This theoretical curve divides the diagram in two parts: above the line, the networks have a density higher than the theoretical critical density n_{c} and should be therefore conducting, whereas below it, the opposite case holds.

Fig. 2c shows a weak deviation of the experimental data with respect to the theory. These results indicate that the critical density for real systems is not perfectly predicted by eqn (1), as some non-percolating networks stand above the line and some percolating networks fall below it.

The fact that the theoretical onset is predicated on the probability of percolation being 0.5 suggests that experimentally some networks could begin to percolate well below that value and some above that threshold may not necessarily be percolating. However the fact that with a small number of samples we were readily able to measure percolating networks close to and below the predicted threshold density indicates a deviation from the theory. This observation however is in agreement with the data derived from Monte Carlo simulations of theoretical networks in Fig. 1d–f. We ascribe this deviation to the existence of imperfections within the experimental network as compared to the ideal networks used to determine eqn (1): the presence of a length distribution, the curvature of the nanowires and a non-isotropy in the angular orientation of the nanowires might play a role. It is now interesting to define the contribution of each of these imperfections on the critical density deviation. At short nanowire lengths the angular distribution clearly dominates resulting in an increase in the percolation threshold. This is likely due to the narrower length distributions experimentally observed for short wires and the dependence of the critical density on the square of the length of the wire as shown in eqn (1). As the length increases the width and influence of the length distribution also increase, resulting in a transition across the theoretical boundary as the longer nanowires start dominating the formation of percolating clusters.

We then generated a network with the three extracted experimental parameters D_{A}, D_{RC} and D_{L}. Such a generated network, which incorporates each of the three imperfections, is reported in the (Fig. S4, ESI†). This allows to mimic fairly well the experimental results as observed by SEM images. Comparison with the generated network suggests qualitatively that the restrictions imposed in the simulations and the premise on which the perturbations were applied to the system are reasonable.

The analysis of Fig. 3 allows us to assess the relative effects of the three imperfections for the considered experimental conditions and AgNW sizes. The dominant parameter is the non-isotropic distribution of AgNW orientations which increases n_{c} of about 40–60%. Then the length distribution can have a certain influence with a decrease of n_{c} of about 20% while the AgNW curvature would only account for a few percentages of n_{c} increase. Therefore if one would consider an experimental method to deposit AgNW in a perfectly isotropic manner, a larger length distribution would lead to a lower critical density. This constitutes an interesting result which calls for further experimental tests. The curvature of the AgNWs appears to play a minor role most likely due to the fact that any reduction in length due to curvature results in an extension of the wire perpendicular to the long axis, therefore effectively increasing the interaction cross section of the wire and increasing the likelihood of cluster formation. The data suggests a slight increase in percolation threshold, which indicates that the shortening effect is not entirely mitigated by the increase in the excluded volume.

Critically this work highlights that the imperfection which has the most impact on the electrical properties of nanowire networks is angular anisotropy. However, this does not appear to be the dominant defect in all systems experimentally. Reduction in the critical density caused by the length distributions appear to start dominating as the nanowire length increases and for wires longer than 20 μm we observe percolating networks below the percolation threshold. The introduction of wire curvatures was found to have a small impact on the network resulting in a moderate increase in the critical density, this is somewhat counter intuitive as the curvature effectively reduces the wire length so one might have expected that it would result in a significant increase. It is likely that the effective loss of length due to the wire curvature is partially mitigated by the increased excluded volume of the wire, increasing the likelihood that the wire will connect with others. These observations are important for future research in 2D networks of 1D objects as it is clear that for optimal network formation the isotropic orientation of the objects is important confirming that spray deposition is preferable over techniques that result in alignment such as meyer rod and spin coating. Additionally this research demonstrates it is not critical to achieve monodisperse object lengths. Finally this work reinforces that the length of the nanowires is likely the most critical factor in the formation of a percolating network and ultimately on the performance of such conductive networks.

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## Footnotes |

† Electronic supplementary information (ESI) available: Matlab scripts for 1D object simulations, extended explanation of imperfection distributions with examples. See DOI: 10.1039/c8nh00066b |

‡ These authors contributed equally to this work. |

This journal is © The Royal Society of Chemistry 2018 |