Closed-Form General Relationship Model for the Interfacial Oxidation of Cylindrically Curved Surfaces

Abstract

A closed-form algebraic expression of the general relationship between film thickness and time during thermal oxidation as attained by Deal and Grove for planar surfaces has remained elusive for surfaces with curvature. Even under a baseline case of constant parameter values to describe oxidant gas-phase transport, diffusion, and reaction, by the conventionally adopted model treating the oxide as fluid capable of flow in accounting for the molecular volume difference between it and the metal from which it was formed and relate the radii describing the oxide relative to those of the metal, numerical integration is required to approximate the time corresponding to any given oxide thickness. Several example sets of such numerical approximation of the relationship between thickness and time by the conventional fluid oxide model on curved cylindrical surfaces are provided here to highlight its lacking closed-form general relationship. In contrast, if instead modeling the oxide as solid and freely expanded from the metal forming it to relate their geometries, it is shown here that a closed-form algebraic expression of the general relationship between oxide thickness and time on cylindrically and in turn spherically curved surfaces is attained in the baseline case of constant parameter values, akin to that preceding by Deal and Grove for planar surfaces. Continuing model refinements will consider dependencies of parameter values on stress state evolving as oxide thickness grows on curved surfaces.