**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Percolation theory

Summary

In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected clusters merge into significantly larger connected, so-called spanning clusters. The applications of percolation theory to materials science and in many other disciplines are discussed here and in the articles Network theory and Percolation (cognitive psychology).
Introduction
A representative question (and the source of the name) is as follows. Assume that some liquid is poured on top of some porous material. Will the liquid be able to make its way from hole to hole and reach the bottom? This physical question is modelled mathematically as a three-dimensional network of n × n × n vertices, usually called "sites", in which the edge or "bonds" between each two neighbors may be open (allowing the liquid through) with probability p

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related people (10)

Related concepts (17)

Percolation

In physics, chemistry, and materials science, percolation () refers to the movement and filtering of fluids through porous materials. It is described by Darcy's law. Broader applications have since

Critical exponent

Critical exponents describe the behavior of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of

Random graph

In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which

Related courses (11)

This course introduces statistical field theory, and uses concepts related to phase transitions to discuss a variety of complex systems (random walks and polymers, disordered systems, combinatorial optimisation, information theory and error correcting codes).

We develop a sophisticated framework for solving problems in discrete mathematics through the use of randomness (i.e., coin flipping). This includes constructing mathematical structures with unexpected (and sometimes paradoxical) properties for which no other methods of construction are known.

Les aménagements hydrauliques sont indispensable pour garantir l'approvisionnement en énergie écophile et renouvelable, de même que l'approvisionnement en eau de bonne qualité et en quantité suffisante pour lutter contre la faim, la pauvreté et les maladies dans le monde.

Related publications (69)

Loading

Loading

Loading

Related units (5)

Related lectures (19)

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.

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.

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.