This paper discusses the development and application of a finite element method for determining the equilibrium shapes of solder joints which are formed during a surface mount reflow process. The potential energy governing the joint formation problem is developed in the form of integrals over the joint surface, which is discretized with the use of finite elements. The spatial variables which define the shape of the surface are expressed in a parametric form involving products of interpolation (blending) functions and element nodal coordinates. The nodal coordinates are determined by employing the minimum potential energy theorem. The method described in this paper is very general and can be employed for those problems involving the formation of three dimensional joints with complex shapes. It is well suited for problems in which the boundary region is not known a priori (e.g., “infinite tinning” problems). Moreover, it enables the user to determine the shape of the joint in parametric form which facilitates meshing for subsequent finite element stress and thermal analyses.
This paper discusses the application of the parametric finite element method for predicting shapes of three-dimensional solder joints. With this method, the surface of the joint is meshed (discretized) with finite elements. The spatial variables (x, y, z) are expanded over each element in terms of products of interpolation (blending) functions expressed in parametric form and element nodal coordinates. The element nodal coordinates which are not constrained by the boundary conditions are determined by minimizing the potential energy function which governs the joint formation problem. This method has been employed successfully in the past to predict the shapes of two dimensional fillet and axisymmetric joints. In this paper, the method is extended to three dimensional problems involving sessile drops formed on a rectangular pad and solder columns formed between two horizontal planes and subject to a vertical force.
An analytical model of solder joint formation during a surface mount reflow process is developed for two-dimensional fillets whose flow may be restricted due to “finite” metallizations on a leadless component and the printed circuit board. Although these height and length constraints on the fillet geometry may result in obtuse contact angles, the solution is obtained in the form of an explicit integral, similar to that previously derived by the authors for the case of acute contact angles. This solution may also be recast into the form of elliptic integrals of the first and second kinds, thereby permitting one to evaluate the fillet geometry using mathematical tables or special function software, if desired, rather than resorting to a computer-based numerical quadrature. In addition an approximate zero-gravity solution is given by means of simple closed-form expressions relating the height, length, contact angles, and cross-sectional area of the fillet. Numerical results generated by implementing the “exact” integral solution for the joint profile are given in the form of dimensionless plots, relating fillet geometry to the solder properties (surface tension and density), amount of solder, chip height, and pad length. Also presented in dimensionless form are the approximate results from the zero-gravity model, which are independent of solder properties, yet are of sufficient accuracy for “small” joints. Because of their dimensionless nature, the results of the present paper may be of maximum utility to process engineers aiming to achieve desired joint geometries (e.g., to maximize fatigue life or to eliminate bridging problems), or to board designers responsible for selecting efficient footprint patterns to maximize board density. Models of solder joint formation, such as the one presented here, may be of most value when used in conjunction with stress analysis packages (e.g., finite element programs) and appropriate fatigue models. In this way an integrated approach to the design of solder joints and circuit boards may be taken, resulting in improved product reliability and performance.
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