Appendix: Capillary Force

At any given point on the fiber-liquid solder-vapor triple line, the local capillary force $\vec{F}$ has magnitude equal to the surface tension $\gamma$, and direction perpendicular to the triple line tangent vector $\vec{d\ell}$. The total force on the fiber is equal to the integral of this quantity around both loops of the triple line TL, so that

$$\vec{F}_{total}=\oint_{TL}\vec{F}d\ell.$$ (2)

The local capillary force vector $\vec{F}$ satisfies the following constraints:

1.

It is at an angle $\theta$ (the contact angle) to the fiber tangent surface, so $\vec{F}\cdot\hat{n}=\gamma\sin\theta$ where $\hat{n}$ is the outward fiber surface normal of unit length.

2.

It is perpendicular to edge tangent vector $\vec{d\ell}$, so $\vec{F}\cdot\vec{d\ell}=0$.

3.

Its magnitude $\vert\vec{F}\vert$ is $\gamma$, the surface tension.

4.

  With the edge integral making a clockwise loop around the fiber, the solder droplet is always on the left, so for $\theta<90^\circ$, $\hat{n}\times\vec{F}\cdot\vec{d\ell}>0$ (<0 for $\theta>90^\circ$).

In cylindrical coordinates transformed so the fiber centerline is the z-axis, its surface will be defined by r=R, so $\hat{r}$ can be used as the surface normal $\hat{n}$. Based on this, we can rewrite the first three above conditions for $\vec{F}$ as three simultaneous equations for the three components of $\vec{F}$ as follows:

$$F_r=\gamma\sin\theta$$ (3)

$$F_\phi d\ell_\phi+F_zd\ell_z=0$$ (4)

$$F_r^2+F_\phi^2+F_z^2=\gamma.$$ (5)

(Note that $\vec{d\ell}$ has no r-component.) We then substitute 3 into 5, and 4 into 5, and solve for $F_\phi^2$and F_z² to give

$$F_\phi^2=\frac{\gamma^2\cos^2\theta d\ell_z^2}{d\ell_\phi^2+... ...\frac{\gamma^2\cos^2\theta d\ell_\phi}{d\ell_\phi^2+d\ell_z^2}$$ (6)

which simplify, satisfying condition 4, to

$$F_\phi=\frac{\gamma\cos\theta d\ell_z}{\vert\vec{d\ell}\vert... ..._z=-\frac{\gamma\cos\theta d\ell_\phi}{\vert\vec{d\ell}\vert}.$$ (7)

The components of $\vec{F}$ given by equations 3 and 7 are integrated using equation 2 to give the capillary force on the fiber.