Perkinson Joy C, Aziz Michael J, Brenner Michael P, Holmes-Cerfon Miranda
Harvard School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138.
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012
Proc Natl Acad Sci U S A. 2016 Oct 11;113(41):11425-11430. doi: 10.1073/pnas.1609315113. Epub 2016 Oct 3.
We propose and experimentally test a method to fabricate patterns of steep, sharp features on surfaces, by exploiting the nonlinear dynamics of uniformly ion-bombarded surfaces. We show via theory, simulation, and experiment that the steepest parts of the surface evolve as one-dimensional curves that move in the normal direction at constant velocity. The curves are a special solution to the nonlinear equations that arises spontaneously whenever the initial patterning on the surface contains slopes larger than a critical value; mathematically they are traveling waves (shocks) that have the special property of being undercompressive. We derive the evolution equation for the curves by considering long-wavelength perturbations to the one-dimensional traveling wave, using the unusual boundary conditions required for an undercompressive shock, and we show this equation accurately describes the evolution of shapes on surfaces, both in simulations and in experiments. Because evolving a collection of one-dimensional curves is fast, this equation gives a computationally efficient and intuitive method for solving the inverse problem of finding the initial surface so the evolution leads to a desired target pattern. We illustrate this method by solving for the initial surface that will produce a lattice of diamonds connected by steep, sharp ridges, and we experimentally demonstrate the evolution of the initial surface into the target pattern.
我们提出并通过实验测试了一种在表面制造陡峭、尖锐特征图案的方法，该方法利用了均匀离子轰击表面的非线性动力学。我们通过理论、模拟和实验表明，表面最陡峭的部分演化为一维曲线，这些曲线以恒定速度沿法线方向移动。这些曲线是非线性方程的一种特殊解，每当表面上的初始图案包含大于临界值的斜率时就会自发出现；从数学上讲，它们是具有欠压缩特殊性质的行波（激波）。我们通过考虑对一维行波的长波长扰动，利用欠压缩激波所需的特殊边界条件，推导出了曲线的演化方程，并表明该方程在模拟和实验中都能准确描述表面形状的演化。由于演化一组一维曲线速度很快，该方程为解决寻找初始表面的逆问题提供了一种计算高效且直观的方法，这样的演化会导致期望的目标图案。我们通过求解能产生由陡峭、尖锐脊连接的菱形晶格的初始表面来说明这种方法，并通过实验证明初始表面向目标图案的演化。
