Describing spatio-temporal features of the Portevin-Le Chatelier effect is known to be a difficult task as they emerge due to collective behavior of dislocations In this article, we describe a dynamical approach to understand three aspects of the phenomenon, namely, the bistable nature of stress envelope, the negative strain rate sensitivity and visualization of dislocation configurations in different regimes of strain rate We shall use some special dynamical techniques such as the reductive perturbative technique and slow manifold approach to understand these features Using a reductive perturbative technique, we derive Stuart-Landau equation for the complex order parameter representing the amplitude and the phase of the limit cycle solution in the neighbourhood of the Hopf bifurcation The slow manifold technique is used to map different parts of the slow manifold with the different branches of the negative SRS More importantly, the method allows us to visualize the dislocation configurations in the regions of the bands

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Self-Organized Criticality

Bistability, negative strain rate sensitivity and visualization of dislocation configurations

Charge transport in disordered insulators and semiconductors is generally agreed to occur by transferring of charge (electrons, holes or ions) betwee n localized states or traps that act as potential wells with respect to the conduction-band level [ 1]. Crucial to the dynamics is the trap occupancy, which is believed to have a significant effec t on the current-voltage characteristics [2]. The filling of the traps at higher voltages (stron g injections) progressively burns the localized states through the medium, thus offering more sites available for the conduction. As a result, the net mobility of the charge carriers is enhanced as their density increases, a phenomenon which relates to the observed dependence of the conductivity on electron density and electron injection in porous nanocrystalline semiconductors [3], as well as electrical degradation and breakdown in insulating polymers [4]. In this paper we propose a model of charge transport in disordered solid materials that is intended to explore the transition to a macroscopically conducting state due to the trap-filling. Our main finding is the complex and multi-scale character of t he transition. Our results can most naturally be explained within a paradigm of self-organized criticality (SOC), perhaps the simplest framework that captures the key signatures of slowly driven evolution processes [5]. Description of the model. Let us consider a hypercubic d-dimensional lattice, where d � 2 is an integer number. Charge can be stored on each node. The lattice nodes are intended to mimic the traps of real insulating materials. In our model only one particle (unit charge) is allowed per node. Adding a particle switches the node from insulating to conducting. Particles are added at random according to the following rule. If the node is empty it absorbs the particle, otherwise excess charge (one particle) is transmitted to a n earest neighboring node. If several neighbors are prone to absorb the particle the destination n ode is a random choice. Tunnelling of charge between the nodes is forbidden. A compensating potential is applied across the lattice in proportion to the number of the absorbed particles. We refer to a setting in which the lattice is confined between two conducting plates exposed to a potentia l difference. The voltage between the plates is increased at a rate so slow that all particles ar e absorbed in traps before a new charge is added. The trap-filling is thus adiabatic and the sy stem evolves through a sequence of quasi-stationary states of progressively increasing trap occupancy. Of interest is a situation when the concentration q of the conducting nodes approaches the

Trap-controlled conduction in disordered materials: Self-organized criticality at play?

In this paper we analyze the sandpile model proposed by Bak, Tang and Wiesenfeld as the canonical example of self-organized critical systems. We find that the sandpile-model can reproduce staircase graphics and also that the distribution of large avalanches recurrence times in this model is log-normal. We also find that the slope of cumulative activity characterize a “province” of generation of avalanches in the same way as the seismic or evolutionary provinces do.

/pdf/on-some-properties-of-the-sandpile-model-of-self-organized-1eyrb969kz.pdf

On Some Properties of the Sandpile Model of Self-Organized Critical Systems Sobre Algunas Propiedades del Modelo de la Pila de los Sistemas Críticamente Auto Organizados

En este articulo se analiza el modelo de la pila de arena propuesto por Bak, Tang y Weinselfed como ejemplo canonico de los sistemas criticamente auto–organizados. Encontramos que la pila de arena puede reproducir graficas tipo escalera, asi como que la distribucion de tiempos de recurrencia en este modelo es log–normal. Hemos tambien encontrado que existe una pendiente caracteristica de la actividad acumulada que caracteriza a una "provincia" de generacion de avalanchas de la misma manera que se hace para provincias sismicas o evolutivas

Self-Organized Criticality

Citations

Bistability, negative strain rate sensitivity and visualization of dislocation configurations

Trap-controlled conduction in disordered materials: Self-organized criticality at play?

On Some Properties of the Sandpile Model of Self-Organized Critical Systems Sobre Algunas Propiedades del Modelo de la Pila de los Sistemas Críticamente Auto Organizados

On Some Properties of the Sandpile Model of Self-Organized Critica Systems

Nonlinear Dynamics and Fractal Avalanches in a Pile of Rice

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