6.4  FEM Calculations of Residual Stress in Single Droplets

Seel developed a FEM approach for the encounter of two islands discussed in the previous section [98][99]. He posed the problem in the same manner as Nix and Clemens and from his simulations he calculated a height Z0   , which minimizes the coalescence energy. By doing so, he imposed a series of displacements in both droplets’ surfaces toward the center and computed the coalescence height which minimizes the energy per unit length of the island surfaces. A sketch is shown in Fig. 6.8.


Figure 6.8.: Structure considered for coalescence simulations. The droplets come in contact at a height Z0   and the surface is displaced at most by Y0   . The angle formed between the droplet and the deposition substrate is identified by θ  . Image based on [98].

A semi-analytical solution for Z0   was also developed in Seel’s work. He took an approach similar to Hoffman, but instead of assuming free energy conservation, he considered the total energy per unit length of the island interfaces, which is given by

E    =  1EY  − (2γ v − γ b)Z ,
 total   2   0     s     g   0

where Z0 = [(r∕sinθ)2 − (r − Y0 )2](1∕2)   . The first term is the elastic energy which Seel obtained by fitting the FEM results in different contact angles. As in the Hoffman model, the second term represents the available energy after the creation of grain boundaries. In contrast to Hoffman’s model, Seel’s analysis does not require energy conservation, because the analysis is restricted to the interface of the islands. However, if the entire system is considered, the energy must be obviously conserved.

The minimum energy point defines the coalescence height Z0   . Although it is possible to obtain an analytical solution for Z0   from (6.8) (dEtotal∕dZ0 = 0  ), it is very cumbersome and a numerical treatment is preferable. Seel’s approach is very flexible, presents results similar to the Freund-Chason model with a slight tendency for underestimation, and is very suitable for engineering purposes.