Three-phase line tensions may become crucial in the adhesion of miero-nano or small droplets on solid planes. In this paper we study for the first time the nonlinear effects in adhesion spanning the full range of physically possible parameters of surface tension, line tension, and droplet size. It is shown that the nonlinear adhesion solution spaces can be characterized into four regions. Within each region the adhesion behaves essentially the same. Especially, inside the characteristic regions with violent nonlinearities, the co-existence of multiple adhesion states for given materials is disclosed. Besides, two common fixed points in the solution space are revealed. These new results are consistent with numerical analysis and experimental observations reported in the literatures.
This paper analyzes the geometric quantities that remain unchanged during parallel mapping （i.e., mapping from a reference curved surface to a parallel surface with identical normal direction）. The second gradient operator, the second class of integral theorems, the Gauss-curvature-based integral theorems, and the core property of parallel mapping are used to derive a series of parallel mapping invariants or geometrically conserved quantities. These include not only local mapping invariants but also global mapping invafiants found to exist both in a curved surface and along curves on the curved surface. The parallel mapping invariants are used to identify important transformations between the reference surface and parallel surfaces. These mapping invariants and transformations have potential applications in geometry, physics, biomechanics, and mechanics in which various dynamic processes occur along or between parallel surfaces.
Based on the concepts of fractal super fibers, the （3, 9＋2）-circle and （9＋2, 3）-circle binary fractal sets are abstracted form such prototypes as wool fibers and human hairs, with the （3）-circle and the （9＋2）-circle fractal sets as subsets. As far as the （9＋2） topological patterns are concerned, the following propositions are proved： The （9＋2） topological patterns accurately exist, but are not unique. Their total number is 9. Among them, only two are allotropes. In other words, among the nine topological patterns, only two are independent （or fundamental）. Besides, we demonstrate that the （3, 9＋2）-circle and （9＋2, 3）-circle fractal sets are golden ones with symmetry breaking.
