A solder ball with diameter ? has volume
When reflown onto two pads without distortion the description of the volume Vbump reads:
Where K is the bond height for the purely spherical bump and p stands for the pad radius. The surface Sbump is given as
When weight is added the bump is compressed in the z direction. For modelling purposes we assume an elliptic model.
The volume of this shape is given as
Where a is defined as
The surface is given as:
Upon insertion of all variables into the formula for Sbump one obtains an expression for the surface energy EdK as function of the deviation dK from the equilibrium bond height K:
Where T0,0 is the surface of the purely spherical bump:
The force constant Fc of this spring is deduced from:
Fc can be derived as:
Where j stands for the ratio p/K of the pad radius p and the bond height K.
When a bump is deformed only in the in-plane direction X by a small amount X. The amount of displacement is taken inversely proportional to the cross section of the bump
The excursion u(z) of the central axis is given as:
where X is the lateral amplitude. It can be shown that the total surface energy Ex can described as the sum of the undisturbed part T0,0 and a contribution due to deflection T2,0 :
From which the force constant for returning to the central equilibrium position can be derived.
When weight is combined with displacement in the horizontal plane we obtain the following result for the total surface energy:
From this formula it can be seen that that is a coupling between the restoring forces in the vertical and horizontal direction. Compared to the other terms T2,1 is very small.
C. van Veen
Read the pdf file here.