Théorie de la percolation | EPFL Graph Search

La théorie de la percolation est une branche de la physique statistique et mathématique qui s'intéresse aux caractéristiques des milieux aléatoires, plus précisément aux ensembles de sommets connectés dans un graphe aléatoire. Cette théorie s'applique notamment en science des matériaux pour formaliser les propriétés d'écoulement dans les milieux poreux et pour la modélisation de phénomènes naturels, comme les incendies. Histoire du modèle L’histoire de la percolation prend ses racines dans l’industrie du charbon. À partir de la révolution industrielle, l’importance économique de cette source d’énergie a stimulé de nombreuses études scientifiques pour en comprendre la composition et en optimiser l’utilisation. Dans les années 1930 et 1940, l’étude qualitative par des méthodes de chimie organique laisse peu à peu la place à des méthodes physico-chimiques plus quantitatives. C’est dans ce contexte qu’est créée en 1938 la (BCURA), association de recherche financée par les p

Critical values in an anisotropic percolation on $\mathbb{Z}^2$

Hao Xue

In this thesis, we consider an anisotropic finite-range bond percolation model on

\mathbb{Z}^2

. On each horizontal layer

\{(x,i):x\in\mathbb{Z}\}

for

i\in\mathbb{Z}

, we have edges

\langle(x,i),(y,i)\rangle

for

1\leq|x-y|\leq N

with

N\in\mathbb{N}

. There are also vertical edges connecting two nearest neighbor vertices on distinct layers

\langle(x,i),(x,i+1)\rangle

for

x,i\in\mathbb{Z}

. On this graph, we consider the following anisotropic percolation model: horizontal edges are open with probability

\lambda/(2N)

with

\lambda\geq 1

, while vertical edges are open with probability

\epsilon

to be suitably tuned as

N

grows to infinity. This question is motivated by a result on the analogous layered ferromagnetic Ising model at mean field critical temperature. We first deal with the critical case when

\lambda=1

. If

\epsilon=\kappa N^{-2/5}

, we see a phase transition in

\kappa

: positive and finite constants

C_1,C_2

exist so that there is no percolation if

\kappaC_2

. The derivation of the scaling limit is inspired by works on the long range contact process. The proof relies on the analysis of the scaling limit of the critical branching random walk that dominates the growth process restricted to each horizontal layer and a careful analysis of the true horizontal growth process, which is interesting by itself. A renormalization argument is used for the percolative regime. We then deal with the supercritical case when

\lambda>1

. If

\epsilon=e^{-\kappa N}

, we can also see a phase transition in

\kappa

. The horizontal and vertical edges can be discovered through subordinate process in each regime. The proof is based on the analysis of supercritical branching random walk but several levels of attritions are introduced to make sure the independent structure. The comparison between our original percolation and the percolation on the inhomogeneous square lattice is used in the renormalization scheme.

EPFL2021

Mechanical Reinforcement of Hydrogels through Physical Crosslinks and Double Network Granular Architecture

Alvaro Lino Boris Charlet

Throughout nature, organisms fabricate a myriad of materials to sustain their lifestyle. Many of the soft materials are composed of water-swollen networks of organic molecules, so-called hydrogels. They generally contribute to the mechanical integrity of the organism, and act as scaffolds for living cells. The ability to fabricate synthetic hydrogels that mimic their natural counterparts would greatly benefit the biomedical field. In particular, synthetic hydrogels have the potential to revolutionize tissue engineering and to enable the fabrication of functional load-bearing soft implants. However, available hydrogels suffer from poor mechanical properties, as they are either too brittle or too soft. While great effort in the field of soft matter has been devoted to the development of hydrogels with improved mechanical properties, they are often not compatible with state-of-the art manufacturing techniques such as additive manufacturing.In this thesis, I present the mechanical reinforcement of hydrogels using two distinctive strategies, and demonstrate their potential as 3D printable materials. I first investigate the use of high functionality crosslinks in metal-coordinated hydrogels. I show that this crosslinking strategy greatly improves the solid-like mechanical properties of viscoelastic gels. I then present the use of hydrogel microparticles to fabricate double network granular hydrogels. I discovered that these materials exhibit an extraordinarily high strength and toughness. Furthermore, the jammed microparticle precursor ink enables the extrusion and 3D printing of this material. This allows the fabrication of hydrogels with locally varying compositions, which can be utilized for example to design stimuli responsive materials. I leverage the granular structure to design recyclable double network granular hydrogels. This is achieved by forming a percolating network that has reversible covalent bonds. I show that this method can be extended to the fabrication of degradable hard plastics. Finally, I conclude by presenting the key findings, and I present a few possible follow-up ideas to further develop the field of load-bearing hydrogels.

EPFL2022

Recycling of Load-Bearing 3D Printable Double Network Granular Hydrogels

Esther Amstad, Alvaro Lino Boris Charlet, Matteo Hirsch, Sanjay Phileas Schreiber

Sustainable materials, such as recyclable polymers, become increasingly important as they are often environmentally friendlier than their one-time-use counterparts. In parallel, the trend toward more customized products demands for fast prototyping methods which allow processing materials into 3D objects that are often only used for a limited amount of time yet, that must be mechanically sufficiently robust to bear significant loads. Soft materials that satisfy the two rather contradictory needs remain to be shown. Here, the authors introduce a material that simultaneously fulfills both requirements, a 3D printable, recyclable double network granular hydrogel (rDNGH). This hydrogel is composed of poly(2-acrylamido-2-methylpropane sulfonic acid) microparticles that are covalently crosslinked through a disulfide-based percolating network. The possibility to independently degrade the percolating network, with no harm to the primary network contained within the microgels, renders the recovery of the microgels efficient. As a result, the recycled material pertains a stiffness and toughness that are similar to those of the pristine material. Importantly, this process can be extended to the fabrication of recyclable hard plastics made of, for example, dried rDNGHs. The authors envision this approach to serve as foundation for a paradigm shift in the design of new sustainable soft materials and plastics.

WILEY-V C H VERLAG GMBH2022

Critical values in an anisotropic percolation on $\mathbb{Z}^2$

Hao Xue

In this thesis, we consider an anisotropic finite-range bond percolation model on

\mathbb{Z}^2

. On each horizontal layer

\{(x,i):x\in\mathbb{Z}\}

for

i\in\mathbb{Z}

, we have edges

\langle(x,i),(y,i)\rangle

for

1\leq|x-y|\leq N

with

N\in\mathbb{N}

. There are also vertical edges connecting two nearest neighbor vertices on distinct layers

\langle(x,i),(x,i+1)\rangle

for

x,i\in\mathbb{Z}

. On this graph, we consider the following anisotropic percolation model: horizontal edges are open with probability

\lambda/(2N)

with

\lambda\geq 1

, while vertical edges are open with probability

\epsilon

to be suitably tuned as

N

grows to infinity. This question is motivated by a result on the analogous layered ferromagnetic Ising model at mean field critical temperature. We first deal with the critical case when

\lambda=1

. If

\epsilon=\kappa N^{-2/5}

, we see a phase transition in

\kappa

: positive and finite constants

C_1,C_2

exist so that there is no percolation if

\kappaC_2

\lambda>1

. If

\epsilon=e^{-\kappa N}

, we can also see a phase transition in

\kappa

EPFL2021

Mechanical Reinforcement of Hydrogels through Physical Crosslinks and Double Network Granular Architecture

Alvaro Lino Boris Charlet

EPFL2022