Dealloying

The Cahn-Hilliard equation on moving surfaces

Dealloying is a current research topic in the field of electrochemistry. In this process the electrochemically more active element of an alloy is dissolved, which results under certain experimental conditions in a nanoporous surface structure. The underlying physical processes for the formation of this structure have not yet been completely understood. For the example of a silver-gold alloy, the authors of the article Evolution of Nanoporosity in Dealloying^[1] suggest a continuous model by a Cahn-Hilirad equation on a moving surface. The motion of the surface in normal direction is caused by the dissolution of silver and the modulus of the velocity is given by

$\displaystyle \operatorname{V}= v_0(c)\left( 1-\delta \operatorname{H}\right),\quad \delta \in \mathbb{R}$

In this equation c is the concentration of gold on the surface, the etching rate v₀ is a decreasing function and H is the mean curvature of the surface. The etching rate satisfies v₀(1)=0, i. e. if the surface is completely covered by gold it is passivated and stops moving. The motion of dissolved gold atoms on the surface is described by the partial differential equation

$\displaystyle \dot{c}+\nabla_\Gamma\cdot\left(b(c)\nabla_\Gamma \left(\gamma \D... ...\cdot\operatorname{V}=c_0\operatorname{V}\cdot\nu , \quad \gamma \in \mathbb{R}$

with

$\displaystyle \Psi (c) = \frac{\theta_{\mbox{\tiny cr}}}{2} c(1-c) + \frac{\the... ...(c)+(1-c)\log(1-c)\right), \quad \theta_{\mbox{\tiny cr}},\theta \in \mathbb{R}$

for the concentration c. On the left side of the equation we have the material derivative of the concentration and the spatial differential operators on the right hand side are tangential derivatitives and the Laplace-Beltrami-operator with respect to the surface. The mobility b is nonnegative and degenerates for c=0 and c=1. Finally c₀ describes the spatial distribution of gold in the bulk.

We perform numerical simulations in order to compare the numerical solutions of these equations to results in electrochemical experiments. The surface is described by a parametric model and the Cahn-Hilliard equation is solved via the evolving surface finite element method^[1]. Additionally results concerning existence of the continuous solution and convergence of the finite element scheme are derived for the case of a given motion of the surface.

The image on the left hand side shows a surface which was computed starting with a planar initial surface and a random initial concentration. In the yellow regions the concentration of gold is almost equal to 1, while it is very small in blue regions. The green portion of the surface is a transition region where the concentration has values between 0 and 1. The complex structure of the surface is evident but in this case it can be partly explained by the random initial concentration.
Looking at the image on the right hand side, the growing complexity of the surface is even more evident. The surface is shown from inside the bulk, i. e. the etching electrolyte is inside the surface. The initial condition for this surface is a small pit not covered by gold, while the rest of the surface is completely passivated. Those portions of the surface can still be seen in the upper middle of the image.Also an unphysical self-intersection can be seen - a situation where the parametric model ceases to describe the evolution of the surface accurately.

Literature

[1] J. Erlebacher, M. J. Aziz, A. Karma, N. Dimitrov and K. Sieradzki, Evolution of Nanoporosity in dealloying, Nature 410 (2001) 450--453

[2] G. Dziuk and C.M. Elliott Finite elements on evolving surfaces, IMA Journal Numerical Analysis 27 (2007) 262--292

[3] C. Eilks and C. M. Elliott: Numerical simulation of dealloying by surface dissolution via the evolving surface finite element method,Preprint Fakultät für Mathematik und Physik, Universität Freiburg, Nr. 08-01 (2008)

author: Carsten Eilks : latest update: 23.4.2008

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