Crystallization waves in thin amorphous layers on heat conducting substrates

Abstract

A crystallization process in thin films is considered, where, driven by the release of the latent heat of fusion, the transformation from an amorphous state to the crystalline state takes place in a progressing wave of invariant shape. The crystallization rate is determined by a rate equation. The influence of the heat loss due to heat conduction into the substrate is taken into account. The resulting system of an ordinary differential equation and an integro-differential equation is solved numerically using a collocation method. The propagation speed of the wave in dependence on a non-dimensional heat loss parameter is determined. It turns out that the existence of a self-sustaining crystallization wave requires the heat loss parameter to be smaller than a certain critical value.A thin film of amorphous material on a semi-infinite substrate is considered, see Figure 1. The system initially has a temperature T S lower than the glass temperature of the film material. If crystallization is triggered (e.g. by localized heating), a transformation from an amorphous state to the crystalline state may take place in a progressing wave of invariant shape. This process is driven by the release of the latent heat of fusion. As the propagation speed in metals, and other materials of practical importance, is rather large, the process is often called "explosive crystallization".A literature survey shows that the quantitative prediction of explosive crystallization is commonly based on semi-empirical laws, e.g. by assuming a temperature dependent propagation speed of the crystallization front, or by introducing an apparent heat transfer coefficient to describe the heat loss to the substrate [1][2][3][4][5].In the present work, the process of self-sustaining explosive crystallization is analysed on the basis of first principles. Fig. 1 A sketch of the problem. k, ρ and c p are the thermal conductivity, density and specific heat capacity, respectively. Subscripts L for layer and S for substrate. Table 1 Relevant quantities (subscripts L: layer, S: substrate)As the crystallizing layer is assumed to be very thin, the energy balance for the layer reduces to the equation of onedimensional heat diffusion with a source term due to the local liberation of latent heat and a heat loss term due to thermal contact with the substrate. The source term is proportional to the crystallization rate, which is determined by a rate equation that has been derived previously [6] on the basis of the crystallization theory due to Kolmogorov [7] and Avrami [8]. Heat conduction in the substrate is described by introducing a continuous distribution of moving heat sources at the interface, based on the single-source solution given in [9]. This leads to an integral representation for the temperature in the substrate in terms of the unknown source distribution.Non-dimensional variables are then introduced to identify the non-dimensional parameters that govern the process, cf. Table 1. Provided the thermal diffusivity of the substrate is much smaller than the...

Influence of heat conducting substrates on explosive crystallization in thin layers

“…for the limiting case of very large activation rate of nuclei as compared to their growth rate [4]. This leads to the following single rate equation as already applied in [8,9]:…”

“…Those theoretical investigations were restricted to adiabatic processes. With regard to heat losses, it was shown in [3,[8][9][10] that a local formulation, such as an effective heat transfer coefficient, is insufficient for describing the effect of heat conduction into the substrate. For a crystallization wave that propagates with invariant properties, the classical solution of a moving heat source was applied and an integrodifferential equation for the temperature distribution in the crystallizing layer was obtained [3,[8][9][10].…”

“…For a crystallization wave that propagates with invariant properties, the classical solution of a moving heat source was applied and an integrodifferential equation for the temperature distribution in the crystallizing layer was obtained [3,[8][9][10]. Together with one [8,9] or more [3,10] rate equations, the integro-differential equation gives a complete set of equations of an eigenvalue problem, with the unknown propagation velocity as the eigenvalue.…”

“…In [3,[8][9][10]] the eigenvalue problem was solved numerically, which is a bit cumbersome owing to the integrodifferential equation. Results were obtained that are of interest from a theoretical point of view as well as with respect to applications.…”

“…Results were obtained that are of interest from a theoretical point of view as well as with respect to applications. Among others, a critical value of a non-dimensional parameter characterizing the heat loss was found in [8,9], leading to dual solutions below the critical value and a lack of solutions above. However, the results were obtained only for a particular set of a rather large number of non-dimensional parameters.…”