Results - Flexible

In order to incorporate bending, first, an analytical approach was pursued to solve for the large deformations along the CNT length due to surface tension. It should be here noted that the angle between the applied force and the CNT -resulting from the interfacial energy between the CNT, the vapor and the liquid- changes with the deformation of the CNT itself. More specifically, the work done by the interfacial energy to deflect the CNT cannot be modeled as resulting from a force with a fixed orientation in the Euclidean space of the beam because the beam undergoes large deformations; instead, the work done must be modeled as resulting from a force that rotates with the induced deformation of the CNT.

From the first two simulations described earlier (Rigid, Slanted), where the CNTs are treated as infinitely rigid beams, we concluded that the triple point mobility is relatively high compared to the mobility at the vapor-liquid interface, and thus the contact angle reaches a steady state in the first few time steps of the simulation. Accordingly, it is assumed in this section that the force has a constant orientation in a coordinate fixed to the tangent of the CNT at the triple point.

The problem of a cantilever beam undergoing large deformations due to a force rotating with the beam rotation has been modeled analytically and can be solved using elliptical functions [1].
The governing differential equations of the cantilever beam can be expressed in the form[2]:
                                                                     Eq.  (1)
with the boundary conditions
                                                                                Eq. (2a)
                                                                                 Eq. (2b)
where E is Young’s modulus, I is the second moment of area, β is the load rotational parameter, L is the length of the beam.

The elastic beam curve can be obtained from (1) and (2) as:
                                              Eq. (3)

The schematic of the configuration is shown below:

The following parameters were defined in order to obtain a non-dimensional solution:
                                                                                                      Eq. (4)
                                                                                                     Eq. (5)
                                                                                       Eq. (6)
                                                                              Eq. (7)

Accordingly, a transformation to the elliptical integral form can be made yielding:
                                                                      Eq. (8)

Subject to the boundary conditions
                                                                                   Eq. (9a)
                                                         Eq. (9b)

Finally, the tip angle α corresponding to a certain load parameter λ can be obtained by integrating Eq. (8) as follows:
                                                                       Eq. (10)
where    is the elliptic integral of the first kind. Finally, the elastic curves can be obtained from:
                                                               Eq. (12)
where
                                                                     Eq. (13)
                                                         Eq. (14)
 is the elliptic equation of the second kind and φ is changing from

Plots describing the effect of force perpendicular to the CNT (contact angle = 90°, e.g. CNT-water ≈ 86°) and an inclined force with respect to the CNT (contact angle = 10°, e.g. CNT-acetone ≈ 9°) are shown in Fig. 1 & 2 respectively. The force parameter λ is varied while the force rotation is kept fixed and the corresponding elastic curves are generated.

                          Figure 1                                            Figure 2         

The video below shows capillary rise of miniscus between two flexible nanotubes that are bending due to surface energy of the miniscus. The m-file needed for this simulation can be downloaded here.