Analytical modeling of isothermal solidification during transient liquid phase (TLP) bonding

Abstract

Transient liquid phase (TLP) bonding has been employed in a range of applications, since it produces joints that have microstructural and hence mechanical properties similar to those properties of the base materials. However, the process generally takes long operational time (from hours to days) since it is controlled by the solute diffusion in solids. Therefore, quantitative prediction of the process kinetics, especially the completion times for different stages (namely heating, dissolution, isothermal solidification and homogenization) in TLP-bonding is very desirable. For those who are interested in the process descriptions and applications in different industries, and the kinetic modeling in general (both analytical and numerical), two detailed reviews [1, 2] are suggested. Examples of the use of numerical and analytical models in determining optimum joining conditions (e.g., bonding temperature and filler metal composition) can be also found in Refs. [1–3]. This communication will focus on the analytical solutions for the isothermal solidification stage during TLPbonding. Although numerical methods provide more accurate prediction of the process kinetics during TLPbonding [2, 3], there is still interest in a quick estimation of the kinetics for certain stages, especially for the isothermal solidification stage. The completion time required for the isothermal solidification stage is generally much longer that the completion time for the previous stages and therefore, a reasonably good estimation of the completion time for isothermal solidification may be used as an approximation for the whole process [1, 2]. This readily explains why much research has been carried out on this particular aspect of the TLPbonding process. When the isothermal solidification stage starts, the liquid phase is at its maximum width (Wmax). The solute build-up in the solid (base metal) is small (Fig. 1) and is, generally, ignored in the analytical modeling of the isothermal solidification stage [2]. The liquid zone shrinks as a result of solute diffusion into the base metal until the joint completely solidifies. It has been pointed out that solute distribution in the liquid can be considered uniform during almost all the isothermal solidification stage [3, 4]; therefore, solute diffusion in the liquid can be ignored. In addition, the base metal can be assumed to be semi-infinite because solute diffusion in the solid is relatively slow. These assumptions make the analytical modeling possible for the isothermal solidification stage [2, 3]. There are many analytical solutions proposed to predict the completion time for isothermal solidification (e.g., Refs [5–12]); however, a close examination of these solutions indicates almost all of them can be classified into two categories. One type has treated the system as two semi-infinite phases with a coupled diffusion-controlled moving solid/liquid interface (Fig. 1), which will be called “two-phase” solutions in this work. The other type has treated the system as a single semi-infinite phase (the base metal) with a constant solute concentration (CαL) at the surface of the base metal (Fig. 1), which will be called “single-phase” solutions. This treatment effectively eliminates the trouble of dealing with the liquid phase or, more importantly, dealing with the MIGRATING solid/liquid interface. Both types of solutions are used extensively in practice. We will examine the derivation procedures of these two types of solutions to study the difference between them. A typical example of the “single-phase” solutions can be found in a paper by Tuah-Poku et al. [12] in a study on TLP-bonding of a Ag/Cu/Ag sandwich joint, which can be also derived from the classical solutions for Fick’s equation (e.g., solutions for semi-infinite media in Ref. [13]). An error function solution is first employed to describe the solute distribution in the semiinfinite base metal with a surface on which the solute