Abstract We use the method of free energy minimization to analyse static meniscus shapes for crystal ribbon growth systems. To account for the possibility of multivalued curves as solutions to the minimization problem, we choose a parametric representation of the meniscus geometry. Using Weierstrass’ form of the Euler-Lagrange equation we derive analytical solutions that provide explicit knowledge on the behavior of the meniscus shapes. Young’s contact angle and Gibbs pinning conditions are analysed and shown to be a consequence of the energy minimization problem with variable end-points. For a given ribbon growth configuration, we find that there can exist multiple static menisci that satisfy the boundary conditions. The stability of these solutions is analysed using second order variations and are found to exhibit saddle node bifurcations. We show that the arc length is a natural representation of a meniscus geometry and provides the complete solution space, not accessible through the classical variational formulation. We provide a range of operating conditions for hydro-statically feasible menisci and illustrate the transition from a stable to spill-over configuration using a simple proof of concept experiment.

中文翻译：

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用于分析带状生长过程中弯液面稳定性的 Weierstras 变分理论

摘要 我们使用自由能最小化方法来分析晶体带生长系统的静态弯月面形状。为了考虑多值曲线作为最小化问题的解决方案的可能性，我们选择弯月面几何的参数表示。使用欧拉-拉格朗日方程的 Weierstrass 形式，我们推导出解析解，提供有关弯月面形状行为的明确知识。杨氏接触角和吉布斯钉扎条件被分析并显示为具有可变端点的能量最小化问题的结果。对于给定的带状生长配置，我们发现可以存在多个满足边界条件的静态弯月面。这些解决方案的稳定性使用二阶变化进行分析，并发现表现出鞍节点分叉。我们表明弧长是弯月面几何的自然表示，并提供了完整的解决方案空间，无法通过经典的变分公式访问。我们为流体静力学可行的半月板提供了一系列操作条件，并使用简单的概念验证实验说明了从稳定配置到溢出配置的转变。