Chemical fracture statistics and universal distribution of extreme values

– When a corrosive solution reaches the limits of a solid sample, a chemical fracture occurs. An analytical theory for the probability of this chemical fracture is proposed and confirmed by extensive numerical experiments on a two dimensional model. This theory follows from the general probability theory of extreme events given by Gumbel. The analytic law differs from the Weibull law commonly used to describe mechanical failures for brittle materials. However a three parameters fit with the Weibull law gives good results, confirming the empirical value of this kind of analysis. Chemical etching of disordered solids is an important technological problem [1], that presents as well interesting questions in the theory of random systems [2]. Strong etching solutions will eventually lead to the fracture of a finite sample, an event defined as " chemical fracture ". In this letter we present a theory for the statistical behavior of this specific fracture mechanism. We show [3] that the deep nature of the chemical fracture statistics relies on the probabilistic theory of extreme events due to Gumbel [4]. Interestingly it is found that our chemical fracture statistics are not practically distinguishable from the Weibull statistics empirically introduced to fit mechanical fracture statistics [5]. The problem is studied using a simple two-dimensional corrosion picture [6] inspired by an experimental study of pit corrosion of aluminum films [7]. The model describes the chemical etching of a random solid by a finite volume of solution. It predicts that the etching process stops spontaneously on a fractal liquid-solid interface as observed experimentally. The two dimensional system is described in Fig. 1: The solid is made of lattice sites exhibiting random " resistances to corrosion " r i ∈ [0, 1] uniformly distributed. It has a width L and a depth Y. At any time t the " etching power " of the solution is proportional to the etchant concentration C(t) : p(t) = ΓC(t). The solution has a volume V and contains an initial number N et (0) of etchant molecules. Therefore p(0) = C(0) = N et (0)/V using Γ = 1. Hereafter we choose p(0) > p c , where p c is the percolation threshold of the lattice. The solution is initially in contact with the solid through the bottom boundary y = 0. At each time-step t, all surface sites with r i < p(t) are dissolved and a particle of etchant …