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Thermomigration in Sn‐Pb solder bumps: Modelling and simulation
Author(s) -
Schuß Stefan,
Hesch Christian,
Weinberg Kerstin
Publication year - 2016
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201610230
Subject(s) - spinodal decomposition , finite element method , temperature gradient , diffusion , boundary value problem , electromigration , mechanics , thermodynamics , materials science , chemistry , phase (matter) , mathematical analysis , physics , mathematics , composite material , organic chemistry , quantum mechanics
It is well known a homogeneous binary alloy subjected to a temperature gradient will become inhomogeneous. This decomposition phenomenon is called thermomigration or thermotransport or Soret effect, which is a cross‐effect in irreversible processes between heat conduction and atomic diffusion. In the present contribution we focus on a diffuse interface model for separation processes affected by temperature gradients coupled with a heat diffusion equation in order to describe thermomigration effects caused by joule‐heating. To be specific, we focus on phase separation in solder bumps consisting of tin and lead. Therefore we discuss the modelling of a Gibbs' configurational free energy density as well as the modelling of the chemical mobility, mobility of thermotransport and Dufour coefficient as sufficiently smooth functions in particle concentration and absolute temperature. The resulting set of partial differential equations involves spatial derivatives of fourth order. Consequently, the variational formulation of the problem mandates approximation functions which are at least C 1 ‐continuous. In order to fulfill this requirement a B‐Spline based finite element scheme is provided. One of the main advantages of B‐Splines is the possibility to represent complex geometries exactly. However, it has been shown that especially curved B‐Spline geometries can not be treated in a straight forward manner in finite element analysis. For this reason we demonstrate the implementation of boundary conditions to avoid the arising numerical perturbations. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)