Capillary forces

Figure 2 shows the dimensionless capillary force on a perfectly rigid fiber parallel to the substrate as a function of fiber position for a set of dimensionless parameters given in the top section of table I. Dimensionless force given in that figure would be multiplied by $\gamma L$ to give the true force. These calculations show that for this pad geometry and parameter set, the horizontal ``aligning'' component of the force is a small fraction of the vertical component, so the fiber will bend toward the substrate.
 
 

Figure 2: Dimensionless capillary force directions and relative magnitudes exerted on a perfectly rigid fiber as a function of fiber position for the system as described by parameters in the top section of table I. The maximum dimensionless force magnitude is 0.151. The fiber is assumed to be aligned parallel to the pad, and height is measured from the fiber axis to the substrate pad.

When the fiber is allowed to bend, the extent of fiber deflection due to capillary force (calculated within Surface Evolver as described above) is significant. For the conditions given in table I, deflection at the free end reaches a maximum of 0.103, which is more than half the fiber diameter. Figure 3 shows the solder surface under such conditions.
 
 

Figure 3: Three views of the calculated solder surface shape around a fiber pulled out of alignment by capillary forces. The grey cylinder shows the original fiber position, and the box is aligned with the fiber to show the angle of fiber deflection relative to the substrate.

For the parameters given here, fiber diameter has the largest effect on displacement: when the diameter increases from 0.2 to 0.3, the maximum deflection falls to just 0.0044, which is 1.5% of the diameter. This is because the elastic energy (equation 1) is considerably more dependent on the fiber diameter than the surface energy.

1998-06-08