Deshpande Paru, Pease Leonard F, Chen Lei, Chou Stephen Y, Russel William B
Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2004 Oct;70(4 Pt 1):041601. doi: 10.1103/PhysRevE.70.041601. Epub 2004 Oct 21.
Cylindrically symmetric structures such as concentric rings and rosettes arise out of thin polymeric films subjected to strong electric fields. Experiments that formed concentric rings and theory capable of explaining these and other cylindrical structures are presented. These rings represent an additional member of a class of structures, including pillars and holes, formed by electrohydrodynamic patterning of thin films, occasionally referred to as lithographically induced self-assembly. Fabrication of a set of concentric rings begins by spin coating a thin poly(methyl methacrylate) film onto a silicon wafer. A mask is superimposed parallel to the film leaving a similarly thin air gap. Electric fields, acting in opposition to surface tension, destabilize the free interface when raised above the glass transition temperature. Central pillars nucleate under small cylindrical protrusions patterned on the mask. Rings then emerge sequentially, with larger systems having as many as 10 fully formed rings. Ring-to-ring spacings and annular widths, typically on the order of a micron, are approximately constant within a concentric cluster. The formation rate is proportional to the viscosity and, consequently, has the expected Williams-Landel-Ferry dependence on temperature. In light of these developments we have undertaken a linear stability analysis in cylindrical coordinates to describe these rings and ringlike structures. The salient feature of this analysis is the use of perturbations that incorporate their radial dependence in terms of Bessel functions as opposed to the traditional sinusoids of Cartesian coordinates. The theory predicts approximately constant ring-to-ring spacings, constant annular widths, and growth rates that agree with experiment. A secondary instability is observed at higher temperatures, which causes the rings to segment into arcs or pillar arrays. The cylindrical theory may be generalized to describe hexagonal pillar/hole packing, gratings, and rosettes with the first being of particular importance given the ubiquitous observation of hexagonal packing. The perturbation analysis presented here is relevant to any system with cylindrical symmetry, for which the radial dependence can be described in terms of Bessel functions.
诸如同心环和玫瑰花结等圆柱对称结构源自受到强电场作用的聚合物薄膜。文中介绍了形成同心环的实验以及能够解释这些和其他圆柱结构的理论。这些环代表了一类结构中的新增成员，这类结构包括柱体和孔洞，是通过薄膜的电流体动力学图案化形成的，有时也被称为光刻诱导自组装。一组同心环的制作首先是在硅片上旋涂一层聚甲基丙烯酸甲酯薄膜。将一个掩膜平行于薄膜放置，留下一个同样薄的气隙。当电场高于玻璃化转变温度时，电场与表面张力相反，会使自由界面失稳。中心柱体在掩膜上图案化的小圆柱突起下成核。然后环依次出现，较大的系统有多达10个完全形成的环。环与环之间的间距和环的宽度通常在微米量级，在一个同心簇内大致恒定。形成速率与粘度成正比，因此具有预期的威廉姆斯 - 兰德尔 - 费里温度依赖性。鉴于这些进展，我们在柱坐标下进行了线性稳定性分析，以描述这些环和类环结构。该分析的显著特点是使用了在贝塞尔函数方面包含其径向依赖性的扰动，而不是笛卡尔坐标中的传统正弦函数。该理论预测的环与环之间的间距、环的宽度以及生长速率与实验结果相符。在较高温度下观察到二次失稳，这会导致环分裂成弧形或柱阵列。圆柱理论可以推广到描述六边形柱体/孔洞排列、光栅和玫瑰花结，鉴于六边形排列的普遍存在，第一种情况尤为重要。这里提出的扰动分析适用于任何具有圆柱对称性的系统，对于这类系统，其径向依赖性可以用贝塞尔函数来描述。
