Acid polishing of lead glass
The authors introduce and discuss a model which accounts for the acid polishing process of lead glass. They consider a molecularly rough surface composed of horizontal layers and submitted to a wet chemical etching process. They introduce the fraction \(\psi _{n}^{j}(t)\) of exposed surface at level \(n\) of species \(j\) and at time \(t\). They prove that the fractions \(\psi _{n}^{j}(t)\) are solutions of the coupled system of ordinary differential equations \( \overset{.}{\psi }_{0}^{j}(t)=-A_{j}\psi _{0}^{j}(t)\), \(\overset{.}{\psi } _{n}^{j}(t)=-A_{j}\psi _{n}^{j}(t)+f_{j}\sum_{k}A_{k}\psi _{n-1}^{k}(t)\), \( n\geq 1\). The initial conditions \(\psi _{0}^{j}(0)=f_{j}\), \(\psi _{n}^{j}(0)=0\), \(n\geq 1\), are added. Here \(f_{j}\) is the fraction of \(j\) molecules. For the resolution of this coupled linear system, the authors use the Laplace transform which leads to an algebraic system whose solution can be expressed in terms of the function \(g(\lambda )=\sum_{k}\frac{ A_{k}f_{k}}{\lambda +A_{k}}\). They then describe some properties of the solution. The paper ends with some considerations on a continuum model, the authors introducing the eigenvectors of the preceding algebraic system and ending with a nonsteady partial differential system.
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