Sharp interface model for eutectic tin/lead

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Sharp interface model for eutectic tin/lead alloy

Collaborator: B. Wagner (FG 1), W. Dreyer (FG 7)

Cooperation with: W.H. Müller (Technische Universität Berlin)

Description:

The reliability of solder joints plays a crucial role in the microelectronic packaging industry. In a real-life situation these solders, such as eutectic SnPb alloy, that consist of a fine mix of tin and lead, begin to form regions of high lead and tin concentration, see Figure 1. This coarsening process which is initiated by diffusion and enforced by the effects of surface tension as well as the exposure to thermo-mechanical stresses, has a drastic influence on the material properties of the solder.

Fig. 1: Lead-rich lamellae after 20 hours of cooling (experiments by W.H. Müller)
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We have investigated here specifically the influence of anisotropic surface tension on the shape of the precipitates formed in this process. For this purpose we derived in [4], on the basis of matched asymptotic analysis, a Sharp Interface Model (SIM) for the two-dimensional phase-field model for SnPb eutectic alloy, as developed and investigated by [1], [2] and [3].

The SIM considers the situation when the solution of the corresponding phase-field model has reached phase equilibrium and focuses on the slow (time scale $\tau= \varepsilon t$ ) evolution of the lead precipitates in a tin matrix. The resulting SIM is solved using a boundary integral method. In this setting it is also possible to derive the exact stationary solutions attained and to determine the dependence of their shape on anisotropy and mass. A linear stability analysis showed that, as in the isotropic case, the stationary solutions are stable.

For the numerical method we used a pseudo-spectral method for the spatial discretization of the boundary integral formulation for the precipitate-matrix interfaces. The parametrization of the boundary integral formulation is chosen such that it allows to isolate the stiffest terms in the evolution equation, which has to be treated implicitly, while the rest is updated explicitly.

Numerical simulations show strong dependence of the inter-precipitate mass flow on the shape of the precipitates.

Fig. 2: Time evolution of two precipitates positioned side by side (left) and on top and bottom (right) with different mass
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Fig. 3: Comparison of the change in area for the side-by-side (black) and top-and-bottom (red) case
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In Figure 2 we start in both cases with precipitates which are stationary solutions by themselves. In both cases the closest distance to each other is the same. We observe that not high curvature but the length of the interfaces with smallest distance plays the dominant role for mass flux, see Figure 3. Moreover, interestingly, the shapes of the precipitates approach the form of elongated lamellae when they gain mass, while they become more circular when they loose mass.

We include the effects of the stress and strain and study their influence on the stability of our solutions.

References:

W. DREYER, W.H. MÜLLER, A study of the coarsening in tin/lead solders, Internat. J. Solids Structures, 37 (2000), pp. 3841-3871.
$\dito$ ,Modeling diffusional coarsening in eutectic tin/lead solders: A quantitative approach, Internat. J. Solids Structures, 38 (2001), pp. 1433-1458.
E. BONETTI, P. COLLI, W. DREYER, G. GILARDI, G. SCHIMPERNA, J. SPREKELS, On a model for phase separation in binary alloys driven by mechanical effects, Preprint no. 1213, Università di Pavia, Istituto di Analisi Numerica del Consiglio Nazionale delle Ricerce, Italy, 2001.
B. WAGNER, W. DREYER, W.H. MÜLLER, Exact solutions of the sharp interface model for eutectic tin/lead alloy, lecture given at: Interfaces and Singularly Perturbed Interface Evolutions, Leipzig, November 29-30, 2002.