A finite element method (FEM) simulation was used to study the elastic behaviour of an
Ag dumbbell structure interacting with a flat substrate (more details in Additional file 1: Figure S4). The model consisted of a dumbbell-like geometry resting on a flat rectangular block. The first case (Figure 3a) describes the earlier stage of dumbbell formation; the length of the adhered part was chosen to be 1 μm long. The second case (Figure 3b) depicts a later stage of dumbbell formation, this website where most of the wire between the balls is detached (the length of the adhered part is 10 nm). In the vicinity of the interface separation edge, the elastic stresses are concentrated and may reach 0.5 to 4 GPa, which can be sufficient to induce interface separation. Note that the stress decreases with the decrease of the length of the adhered part; thus, only
relatively Selleckchem Roxadustat short NDs are able to detach from the substrate completely. Figure 3 FEM simulations of elastic behavior of a ND adhered to a substrate. The bulb radius is 175 nm, total wire length 2 μm, and the wire cross section is pentagonal of 100 nm in diameter. (a) First case – adhered part length 1 μm. (b) Second case – adhered part length 10 nm. The thermal stresses induced by contraction of the NW due to cooling may play a significant role in the interface separation as well. The thermal strain th can be estimated from the following equation: (2) where α Ag is the thermal expansion coefficient of silver and ΔT is the difference of the initial and final temperatures. The thermal expansion coefficient
of bulk silver is 19.7 × 10-6/K , and considering the temperature difference of 680 K, the strain for such a process is approximately 1.34%. Calculating the thermal stress by σ th = E Ag th, where E is Young’s modulus for silver (E Ag ≈ 83 GPa), one yields σ th ≈ 1.1 GPa. As the result of superposition of the elastic stress of bent NW and thermal stress, interface separation takes place similarly to crack propagation. Contact area and static friction The contact area, as well as ZD1839 friction between the end bulbs and the substrate, will strongly depend on the shape of the bulbs. According to the experimental observations, the end bulbs of the NDs have an ellipsoidal shape that is close to prolate spheroid with the semi-axes R 1 and R 2. For purposes of simplicity, we will use spherical ball approximation, justified by the ratio R 1/R 2 ~ 1. Thus, the effective radius will only be used. The real shape of the bulb is a result of the dynamic interplay of surface tension and adhesion forces in a liquid droplet followed by solidification. In this regard, two boundary cases can be considered.