Fractals, Vol. 18, No. 4 (2010) 461–476 c  World Scientific Publishing Company DOI: 10.1142/S0218348X10005032 KINETIC RESPONSE OF SURFACES DEFINED BY FINITE FRACTALS PRADEEP R. NAIR ∗ and MUHAMMAD A. ALAM † School of Electrical and Computer Engineering Purdue University, West Lafayette, IN 47907, USA ∗ pnair@purdue.edu † alam@purdue.edu Received December 28, 2009 Accepted April 30, 2010 Abstract Historically, fractal analysis has been remarkably successful in describing wide ranging kinetic processes on (idealized) scale invariant objects in terms of elegantly simple universal scal- ing laws. However, as nanostructured materials find increasing applications in energy storage, energy conversion, healthcare, etc., one must reexamine the premise of traditional fractal scaling laws as it only applies to physically unrealistic infinite systems, while all natural/engineered systems are necessarily finite. In this article, we address the consequences of the ‘finite-size’ problem in the context of time dependent diffusion towards fractal surfaces via the novel tech- nique of Cantor-transforms to (i) illustrate how finiteness modifies its classical scaling expo- nents; (ii) establish that for finite systems, the diffusion-limited reaction is decelerated below a critical dimension D ∗ F and accelerated above it; and (iii) to identify the crossover size-limits beyond which a finite system can be considered (practically) infinite and redefine the very notion of ‘finiteness’ of fractals in terms of its kinetic response. Our results have broad implica- tions regarding dynamics of systems defined by the same fractal dimension, but differentiated by degree of scaling iteration or morphogenesis, e.g. variation in lung capacity between a child and adult. Keywords : Finite Fractals; Clusters; Biosensors; Diffusion Limited Aggregation; Morphology. 461