Can one ‘‘hear’’ the thermodynamics of a (rough) colloid?

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Abstract
We calculate the effects of surface singularities, e.g., wedges, on the thermodynamics of electric double layers near a rough colloid or charged membrane. Each wedge yields an additive contribution to the free energy scrF that is a function of the angle. We give a geometrical analysis of scrF entirely similar to that of vibration eigenmodes given by Kac in ‘‘Can one hear the shape of a drum?,’’ with a probabilistic Brownian representation of scrF, our analysis yields a universal scaling law for the free energy of a fractal colloid with its Minkowski dimension. The case of a colloidal random walk is derived from conformal invariance.