Liquid solder capillary force

While the solder droplet is molten, the force on the fiber is assumed to arise from surface tension and hydrostatic pressure in the liquid solder. These forces are determined by the shape of the solder-fiber-air triple line as described in the appendix, which in turn depends on the shape of the solder droplet. The droplet is assumed to reach its minimum-energy shape, which is calculated using the Surface Evolver program written by Ken Brakke [1,2]. This program moves the surface to minimize the total energy of the system, including energies corresponding to the solder-air, solder-fiber, and solder-substrate surfaces, and also the pressure energy. Gravitational energy is neglected here due to the small droplet size. Surface Evolver creates a triangular mesh of the relevant surfaces, and moves the vertices of the mesh down the gradient of total system energy while maintaining constant solder volume and satisfying geometrical constraints. It also handles all mesh refinement and equiangulation, so the program output is the Delaunay triangulation of a very close approximation to the minimum-energy shape at the specified resolution.

A dimensionless formulation is employed with parameters as shown in table I. Dimensionless parameters are given in reference to the lengthscale L equal to half of the true pad width, and the solder-air surface energy $\gamma$, so that for example, the true fiber radius is 0.1L and the true modulus is $E\gamma/L$.

  

Table I: Dimensionless parameters used in Evolver calculations

Parameter	Symbol	Value
Pad length and width		2
Fiber radius	R	0.1
Solder droplet volume		1
Solder-fiber surface energy	$\gamma_{SF}$	$\cos(50^\circ)$
Solder-pad surface energy	$\gamma_{SS}$	$\cos(30^\circ)$
Fiber modulus	E	107
Fiber fixed end-droplet center distance	D	7.3
Fiber free end-droplet center distance		5.8

The region of the substrate outside of the square contact pad is taken to be perfectly non-wetting, so the contact angle of the solder is equal to 30$^\circ$ on the corners, and greater than 30$^\circ$ along the edges where the droplet is constrained by the edge of the pad.