CAN ONE HEAR THE THERMODYNAMICS OF A (ROUGH) COLLOID

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 F that is a function of the angle. With a probabilistic Brownian representation of F, we give a geometrical analysis of F entirely similar to that of vibration eigenmodes given by Kac in "Can one hear the shape of a drum?." 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.