Electrical breakdown in disordered solids

In this chapter we shall begin with the problems associated with failures in disordered solids under the influence of an electric field, because this kind of failure is simpler than the mechanical failure or fracture. However, all the cases of failure present some common features, so that the study of electric failures will provide a convenient framework for introducing several important concepts. 

We shall consider two extreme kinds of systems. In the first kind, the system is a conductor and by application of a voltage between two electrodes (for the sake of simplicity the two electrodes will be taken parallel) a current flows from one electrode to another. The failure occurs when the current density becomes larger than a threshold value. Consequently, the system becomes nonconducting. The system behaves exactly as a fuse which is destroyed when the current is too large. We shall call this failure the fuse failure. In the second case, the system is a perfect insulator and a voltage is applied between the two electrodes. Again, beyond a definite (threshold) value of the electric field, the system breaks down and becomes conducting. This phenomenon is well-known in the physics of dielectrics, since it limits the application of dielectrics as insulators. We shall talk about the dielectric problem for this kind of failure. 

In the present work, we shall not discuss the exact nature of the failure, i.e. its microscopic mechanism. In the fuse problem, the mechanism of the failure is very well-known (it is merely the Joule effect), but in the dielectric problem the mechanism is much more complicated (O Dwyer 1973). The reason is that we intend to attack the problem from a point of view which is of tremendous importance for statistical analysis. If the sample is perfectly homogeneous the failure will take place in all the portions of the sample. In the first case the current density is uniform in the sample and in the second, the electric field is the same everywhere. If the threshold value is reached, the failure will be general and the sample will explode. In fact, this never happens. The failure always begins as a local event and progressively becomes general. This is because there are weak points in the system. The failure always begins at these weak points. The existence of weak points is due to the fact that solids are never homogeneous. This means that the 

The problem of the infiuence of disorder on the physical properties of materials is very important since it has implications in many fields composite materials, polymers, emulsions, ionosphere, etc. The concept of disorder is very general and in each case it is necessary to state precisely what we mean by disorder. 

The first thing we have to specify is the length scale of the disorder. In the present case, since we are not interested in the exact mechanism of the failures, the length scale will be larger than the regions in which the local failure appears. For this reason we shall assume that the defects (consequence of the disorder) are macroscopic. It is difficult to always define an exact length scale but it will be assumed to be much larger than the atomic distances. 

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