Model

Capillary rise of vapor front in between two nanotubes can be modelled as interface migration involving triple junctions described in the work of Sun et al [1]. The schematic of a triple junction involving Nanotube, Vapor, and Liquid is shown below.

The weak statment goverining interface migration and triple junction movement is given by:

where is the velocity of triple junction, is the virtual motion of triple junction, is the mobility of the triple junction, is the normal veolocity of the interface, is the virtual velocity of the interface and is the mobility of the interface. Assuming at equilibrium, the junction mobility is infinitely large, the first term of above equation tends to zero.

The virtual motion of the nodal points is given by:

where through are shape functions given by:

where is the distance measured from the middle of the element and is the length of the element.

Similarly, the interface velocity is given by:

Total energy for an element varies as:

where is the surface energy and is energy associated with phase change. This can be accounted to virtual motions of the nodes by:

Using above equations, the weak statement can be written as:

where is:

is:

and is the vector of nodal velocities of a single element. For a multi-element system, the H matrices and f matrices of each element should be combined to yield global H and f matrices. For our project, the forces due to surface energies at the triple point should also be added to appropriate element of the f matrix to complete the model.

Once the model is solved for nodal velocities, numerical approach can be used to update the nodal positions and evolve the structure.