Fuse model

For the fuse problems with various other criteria of failure, for example breakdown due to local Joule heating in a random thermal fuse model, see Sornette (1987) and Sornette and Vanneste (1992) and also Section 2.2.6. 

We give here a brief resume of the theoretical results of the failure behaviour in the fuse model. 

The shortest path and the electromigration fuse model We introduce in this section the notion of the shortest path in a resistor lattice, which is a pure geometrical notion. We shall see that some cases of failure can be described by this geometrical picture. In this section we present also a variant of the fuse model, the electromigration model. The interest of the notion of shortest path is that it can be used in the case of the dielectric breakdown. In the following part of this chapter (Section 2.3 on dielectric breakdown) we shall describe a dielectric model and its experimental realisation which are well analysed with the help of the shortest path approach. 

The electromigration fuse model of Bradley and Wu (1994) is a modification of the fuse model discussed above. As in the fuse model, the electromigration model was studied on a square lattice of resistors. The criterion for a resistor to break is not some value of the current (as in the fuse model) but the total charge that crossed this resistor from the time of application of the voltage. For a constant current Iq the whole sample breaks after a time r. The problem is to determine r for given value of p. For a single bond, the time of failure ti is given by... 

If p is very near to 1, the mean time (r) is related to the longest probable defect with size ric. As in the fuse model, is given by (2.10). This means that the bonds which fuse are also the bonds with largest current. 

We have four quantities which go to zero at Pc with the same exponent the shortest path, the failure current If and the number of broken bonds iVf in the fuse model and the failure time (r) in the electromigration fuse model, and one can ask if this similarity has a deeper reason. Although there is only numerical evidence concerning the exponent of iVf, we admit that it is effectively the correlation length exponent. Concerning (r), its proportionality with g becomes better and better when approaching Pc-... 

In conclusion, the two criteria, namely that the current in a link or that the temperature of a singly-connected bond reaches a specific value at the failure, give the same results. It was in fact expected, since the two quantities ATgc and ii, are related by the relation (2.43). We shall see below the experimental results giving support for the second approach and making the fuse model a realistic one. 

We shall use the same machinery as that developed above in the case of the lattice fuse model. In order to mimic the insulator part, one takes capacitors with a given value Ci as insulating bonds. For the conducting bonds, one takes very large capacitor Cq such that Co Ci (Beale and Duxbury 1988). By this choice the voltage across a Cq capacitor is zero, as within the conductor the field must be zero, p is the fraction of Cq capacitors which are now seen as defects. Of course, the other possibility is to take... 

The distribution function F E) giving the probability that the sample will break when a field E is applied is obtained by a procedure identical to that we used for the fuse model. The result is... 

We mentioned this concept when we introduced the electromigration fuse model (Section 2.2.5). We recall it in the present context of the dielectric breakdown problem. Above we considered a walker which jumps from bond to bond since the problem was to break bonds until failure. Here, we adopt a slightly different definition specific to the present problem. 

The interest of the concept of shortest path is evident in the case of the dielectric breakdown. In the electromigration fuse model, it was useful only in two dimensions, but here it can be of interest in all dimensions. 

Figure 2-118, Cross-section of the 3D model of formic add (HCOOH), The van der Waals radius of each atom of the molecule is taken and by fusing the spheres the van der Waals surface is...

In addition to electrophilic attack on the pyrrole ring in indole, there is the possibility for additions to the fused benzene ring. First examine the highest-occupied molecular orbital (HOMO) of indole. Which atoms contribute the most What should be the favored position for electrophilic attack Next, compare the energies of the various protonated forms of indole (C protonated only). These serve as models for adducts formed upon electrophilic addition. Which carbon on the pyrrole ring (C2 or C3) is favored for protonation Is this the same as the preference in pyrrole itself (see Chapter 15, Problem 2)1 If not, try to explain why not. Which of the carbons on the benzene ring is most susceptible to protonation Rationalize your result based on what you know about the reactivity of substituted benzenes toward electrophiles. Are any of the benzene carbons as reactive as the most reactive pyrrole carbon Explain. 

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