The influence of stress on spiral growth

Abstract

A differential equation and boundary conditions describing rotating spirals of stationary shape are presented. If the dislocation stress field is strong enough to allow the formation of hollow dislocation cores (radius: r hc) both the analytical approximation as the phase plane analysis of the differential equation indicate the occurrence of three kinds of spiral: (i) global spirals, running from the dislocation centre ( r =0) to infinity, (ii) inner spirals, running from r =0 to r hc, and (iii) outer spirals running from r = r hc to r = ∞. If the hollow core is kinetically and thermodynamically possible, only types (ii) and (iii) occur. From the solution of the differential equation, the angular velocity ω 1, and thus the spiral growth or dissolution rate, is obtained numerically. The ω 1 values indicate the sudden occurrence of steep and macroscopic etch pits below a certain critical undersaturation. Also it is found that stress opposes growth slightly but favours dissolution considerably. Starting at high undersaturation, the shape of a spiral at decreasing undersaturation is as follows: closely spaced dissolution spiral, hollow core inner and outer dissolution spiral, hollow core inner and outer growth spiral, growth spiral with depressed centre en finally widely spaced archimedean-type growth spiral. Some observations of the above mentioned spiral types are shown.