Non-equilibrium statistical theory of water treeing in polymeric cable insulators

Abstract

A non-equilibrium statistical theory of water treeing in polymeric cable insulators, which treats water tree growth as a stochastic process, is presented. In this treatment the deterministic equation for the rate of water tree growth is made stochastic by the addition of a fluctuation term. The fluctuations are used to model the effect of the complex topologically connected microstructure of the polymeric insulator on water tree growth. Such considerations lead to a generalized Langevin equation for the water tree's growth rate as well as an equivalent Fokker - Planck equation for the probability density distribution of the water tree length. Many of the major attributes of water tree growth are shown to be a natural consequence of this equation. The self-affine fractal object for water tree morphology is first constructed, based both on the self-affinity of the time-correlating fluctuations and on the scale-invariance of the fundamental dynamic equation dominating water tree growth. The empirical two-parameter Weibull distribution of water tree length in the literature is also derived. Good quantitative agreement between theory and previously reported experimental results is shown.