Kirkendall Effect in Creating Three-Dimensional Metal Catalysts for Hierarchically Porous Ultrathin Graphite with Unique Properties

In this work, we report an innovative mechanism, the Kirkendall effect, in creating three-dimensional (3D) microporous catalysts with tunable pore sizes for the growth of hierarchic ultrathin graphite foams (HP-UGFs) with unique properties. Employing the Kirkendall effect is one of the first demonstrated for fabricating 3D porous catalysts, where tunable pores of 1.9–8.3 μm are created on 3D interconnected struts (∼100 μm). With the catalysts, we readily synthesized freestanding HP-UGFs that offer higher crystallinity and electric conductivity, larger surface area, as well as enhanced electric invariance to strains compared to those of conventional ultrathin graphite foams. A gauge factor as low as ∼10 at a strain as high as 80% is achieved owing to the unique porous corrugations created on the microstruts of the HP-UGFs. This work may inspire a new paradigm in designing and synthesizing a new type of 3D porous architecture made of 2D materials with controlled local corrugations, which could greatly benefit flexible electronics

Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations

We compute the Lie symmetry algebra of the equation of Helfrich surfaces and we show that it is the algebra of conformal vector fields of R2. We also show that in the particular case of the Willmore surfaces we have to add the homothety vector field of R3 to the aforementioned algebra. We prove that a Helfrich surface that is invariant w.r.t. a conformal symmetry is a helicoid and that all such surface solutions satisfy one and the same system of ordinary differential equations obtained by symmetry reduction. We also show that for the Willmore surface shape equation the symmetry reduction leads to two systems of ODEs. Then we construct explicit solutions in the case of revolution surfaces. The results obtained can be extended to the study of PDE problems in 2 spatial dimensions admitting conformal Lie symmetries