Kinetics of Simultaneous Transformations

It is quite frequent for a number of solid–state transformations to occur concurrently, starting from the same parent phase. The transformations may occur at different rates, but the resulting competition for space or for the partitioning of driving force between the precipitating phases can be seminal to the development of many microstructures found in commercial alloys. This paper reviews the overall transformation theory available for dealing with simultaneous transformations. The theory discussed is generally applicable, but is illustrated with specific reference to secondary hardening steels and structural steels. Introduction There are at least three circumstances in which a phase might transform into more than one product: 1. The equilibrium precipitate may be difficult to nucleate. Consequently, decomposition starts with the formation of one or more metastable phases which are kinetically favoured. These must eventually dissolve as equilibrium is approached. There are classic examples of this in the age–hardening of aluminium alloys and in secondary hardening steels. A recent example is the crystallisation of metallic glass, at first to a metastable phase [1]. In each case, the formation of the metastable phase is accompanied by a reduction in free energy causing an exaggerated retardation of the stable phase. 2. All of the product phases may be at equilibrium; e.g. the transformation of austenite into a mixture of ferrite and graphite. However, the transformation products do not grow in a coupled manner and may be sufficiently separated in time to be treated as sequential rather than simultaneous. 3. The product phases may be coupled as in the formation of pearlite, with a common transformation front. It is the first case which forms the subject of this review. Avrami Theory A model for a single transformation begins with the calculation of the nucleation and growth rates using classical theory, but an estimation of the volume fraction requires impingement between particles to be taken into account. This is generally done using the extended volume concept of Johnson, Mehl, Avrami, and Kolmogorov [e.g. 2] as illustrated in Fig. 1 (henceforth referred to as “Avrami theory”). Suppose that two particles exist at time t; a small interval δt later, new regions marked a, b, c & d are formed assuming that they are able to grow unrestricted in extended space whether or not the region into which they grow is already transformed. However, only those components of a, b, c & d which lie in previously untransformed matrix can contribute to a change in the real volume of the product phase (identified by the subscript ‘1’) : dV1 = (1 − V1 V )dV e 1 (1)