CONNECTED LENGTH OF FRACTALS (WHY ARE NUCLEATION AND CRITICAL RELAXATION STEPWISE, WHY POLYMERS DESCRIBE n = 0 SPINS)

Abstract

A stepwise "growth" process, which divides a fractal cluster into branches of sequentially connected particles, helps us to show the following: A reversible growth of a cluster is dominated by an addition of subunits to (or subtraction from) the branches' ends. Main contribution is due to the smallest subunits, viz. single particles. Two consequences follow. First, large scale aggregation and fragmentation may be safely ignored, which supports the classical nucleation theory. Second, relaxation of a cluster is described by the number of back and forth steps needed to traverse the branches' typical length, which supports a recent theory of critical slowing-down. The growth process also helps us to show that bifurcation is due to a fluctuation, modified by a mutual screening, of the branches' ends. For thermal critical clusters, the screening is determined by two opposing effects: Repulsion due to excluded volume, and attraction due to an "induced-growth" of ends by ends. The resultant determines the clusters' deviation from ideality. However, in the case of the uniformly oriented n = 0 vector spins (with no up-or-down option), there can be no fluctuation and clusters grow without bifurcation. This "explains" why linear real polymers describe critical clusters of n = 0 spins.