Pattern formation during electropolishing

Using atomic force microscopy, we find that the surface morphology of a dissolving aluminum anode in a commercial electropolishing electrolyte can exhibit both highly regular and randomly packed stripe and hexagonal patterns with amplitudes of about 5 nm and wavelengths of 100 nm. The driving instability of this pattern formation phenomenon is proposed to be the preferential adsorption of polar or polarizable organic molecules on surface ridges where the contorted double layer produces a higher electric potential gradient. The enhanced relative coverage shields the anode and induces a smaller dissolution rate at the ridges. The instability is balanced by surface diffusion of the adsorbate to yield a length scale of $4\ensuremath{\pi}{(D}_{s}{/k}_{d}{)}^{1/2}$, where ${D}_{s}$ is the surface diffusivity and ${k}_{d}$ is the desorption coefficient of the adsorbate, which correlates well with the measured wavelength. A long-wavelength expansion of the double-layer field yields an interface evolution equation that reproduces all of the observed patterns. In particular, bifurcation analysis and numerical simulation yield a single voltage-dependent dimensionless parameter $\ensuremath{\xi}$ that measures a balance between smoothing of adsorbate concentration by electric-field-dependent surface diffusion and fluctuation due to interfacial curvature and stretching. Randomly oriented stripes are favored at large $\ensuremath{\xi}$ (low voltage), while random hills dominate at small $\ensuremath{\xi}$ (high voltage) with perfectly periodic stripes and hexagonal hill patterns within a small window near $\ensuremath{\xi}=1.$ These predictions are in qualitative and quantitative agreement with our measurements.