Theory on Bending in Cantilever Beams With Adsorbed Islands

Traction between adsorbed islands and the substrate is commonly seen in both living and material systems: deposited material gathers into islands at the early stage of polycrystalline film deposition and generates stress due to lattice mismatch, cells exert cellular traction to extracellular matrix to probe their surrounding microenvironment in vivo, and so on. The traction between these islands and the substrate can result in perceivable macroscopic deformation in the substrate and may be measurable if the substrate is a cantilever beam. However, currently broadly used Stoney equation is incapable of handling such boundary condition. In this paper, we give the closed-form expression on the resulted curvature in substrate beams by distributed tractions. Such a relationship could be employed to monitor the stress evolution during thin film deposition, to quantify the stress level of cell traction as cells adhere to cantilever beams, and other related mechanical systems like charging–discharging induced stress in island-patterned electrode films. Moreover, we found that follower traction induced by an array of islands could lead to negative curvature. It shields light on the early stage compressive stress during polycrystalline film deposition.