COSYSLISISNous rejoindreStages Reaseach internship offer (2019) : towards a more realistic modelling of carbon nanotube percolating networks : from random graphs to the resistance of water quality sensors Afficher le menu principal

Reaseach internship offer (2019) : towards a more realistic modelling of carbon nanotube percolating networks : from random graphs to the resistance of water quality sensors

Location : LPICM, (Ecole Polytechnique), PLATINE platform and / or CERMICS (ENPC)


The Laboratory of Interfaces and Thin Films, and more specially PLATINE team, is developing water quality nanosensors based on percolating networks of carbon nanotubes (CNTs). The idea of these sensors is to measure a variation of the resistance of the CNTs network when it is immersed in water, with an increasing concentration of a target ion. The electronic properties of CNTs are indeed highly sensitive to their environment. They could even be « tunable » in a selective way. By functionalizing CNTs with specific polymers, we are looking for predictable variation of the resistance of the sensors in water, under increasing concentration of specific target ions.

These CNTs percolating networks involve many conduction pathways, each including not only linear portions of nanotubes, but also contacts between nanotubes (and contacts between metallic electrodes and nanotubes at each path departure (see Figure 1)). The understanding of all the contributions to the total resistance, due respectively to the linear resistances, the CNT/CNT contact resistances, and the metallic electrode/nanotube contact resistances, is not complete yet. Neither is its dependence on the main geometrical parameters describing the network.

The question of to what extent is the equivalent resistance dominated by contacts resistances between CNTs (or by the linear resistance along tubes), depending on the geometry, hasn’t been thoroughly answered yet. As the response of a sensor to a varying concentration of a target ion in water depends on all the contributions to the total resistance, the latter have to be understood in detail (if possible for all possible device geometries).

Proposed research

The goal of this internship is thus to continue the modeling of percolating networks of CNTs, following the work done by Robert Benda at the start of his PhD [1] (september to december 2018), extending it to more realistic geometries, understanding systematically the contributions of the different types of resistances and the influence of geometry.

This work [1] ended up with a method to estimate the contribution of the CNT/CNT contact resistance vs. linear resistance on the overall resistance, for test model networks. Many improvements, in the continuity of this work, can be considered :

- The Python code developed to generate randomly 2D percolating networks of ‘stick’ (rigid) CNTs and to compute their electrical resistance with the adjacency matrix method [2] has to be merged with a previous code done by master students [3] to analyse microscopy images of percolating networks (with Image J software), which extracted the distributions of the main geometrical parameters (length, waviness, orientations of CNTs) and generated a percolating network reproducing these statistical distributions (hence, more realistic).

- The possibility for CNTs to be curved (and not only ‘stick’) has to be implemented in the code [1], and the possible effects on the computed resistance estimated.

- More systematically, the influence of the statistical law chosen for a given geometrical parameter (as, for instance, the orientation of the tubes) has to be understood, in the spirit of article [4] (Simoneau and al.). The dependence of the results on other modeling choices, such as the non-penetrability of CNTs (hard-code soft-shell model, see [5]), will have to be analysed to assert the robustness of the approach.

- Given the need to perform numerous simulations, the improvement of the complexity of the code and its parallelization could also be adressed,as well as other mathematical problems related to percolation theory and the underlying random variables [6].

Team & organization

Position located in greater Paris area : Ecole polytechnique, Route de Saclay, 91128 Palaiseau, France.

NACRE research team (IFSTTAR, Ecole polytechnique, CNRS), in Platine research platform ( in LPICM laboratory (

The team counts 3 researchers, 2 engineers, 1 technician, 8 to 10 young researchers (interns, PhDs, postdocs) and 2 startup projects. One of these startups (in creation) notably focuses on the commercialization of the sensor array technology for water monitoring.

and /or

CERMICS (ENPC) Batiment Coriolis, 6 et 8 avenue Blaise Pascal, Cité Descartes – Champs sur Marne, 77455 Marne la Vallée Cedex 2.

Application and deadlines

The successful candidate will have a strong background in coding (if possible in Python), numerical methods, and basic knowledge in linear algebra. Interests both in the physical aspects of the problem and on the mathematical underlying subtleties are welcome.

Fluent English (written and spoken) if possible, high proficiency in technical writing and presentations. Autonomy, spirit of initiative, decision-making skills, creativity, team working. Project management skills.

To be considered, applications must include detailed resume and motivation letter.

Starting date: Adjustable. Preferably from February or March 2019.


Robert Benda (robert.benda(at)

Bérengère Lebental (berengere.lebental(at)


[1] Modelling of 2D 'stick' carbon nanotube percolating networks, Robert Benda, December 2018 (available on demand at robert.benda(at) for students interested by the offer).

[2] F. Y. Wu, Theory of resistor networks: the two-point resistance, Journal of Physics A : Mathematical and General 37 (2004), 6653-6673.

[3] Modélisation de la piézorésistance d’un réseau de nanotubes de carbone : Analyse d’image et simulation numérique, Projet de recherche en laboratoire, Hippolyte Verdier, Clément Verlhac.

[4] Louis-Philippe Simoneau, Jérémie Villeneuve, Carla M. Aguirre, Richard Martel, Patrick Desjardins, and Alain Rochefort, Influence of statistical distributions on the electrical properties of disordered and aligned carbon nanotube networks , Journal of Applied Physics 114, 114312 (2013).

[5] Louis-Philippe Simoneau, Jérémie Villeneuve, and Alain Rochefort, Electron percolation in realistic models of carbon nanotube networks, Journal of Applied Physics 118, 124309 (2015).

[6] Nicolas Curien, Random Walks and graphs, 2018.


Figure 1. Example of a 2D percolating network of CNTs generated for N=200 CNTs of length 5 (the electrodes being at y=0 and y=20)