The mechanical and chemical properties of soft solids are crucial to many applications in biology and surface science. Recent studies use wetting by liquid drops to probe the surface mechanics of reticulated polymer networks, leading to controversial interpretations. This controversy relates to the long-standing paradox of Young’s law for the liquid contact angle, which invokes only a horizontal force balance. Recent work shows that, for very soft materials, the solid’s surface tension plays a key role for the vertical force balance, involving a singular ridgelike deformation exactly at the point where the droplet pulls on the network. A hotly debated question is whether unexpected measurements on this singular deformation can be attributed to nonlinear bulk elasticity or whether these provide evidence for an intricate surface elasticity, known as the Shuttleworth effect. Here, we theoretically reveal the nature of the elastocapillary singularity on a hyperelastic substrate with various constitutive relations for the interfacial energy. First, we finely resolve the vicinity of the singularity using goal-adaptive finite-element simulations. This simulation confirms that bulk elasticity cannot affect the force balance at the contact line. Subsequently, we derive exact solutions of nonlinear elasticity that describe the singularity analytically. These solutions are in perfect agreement with numerics and show that both the angles and stretch at the contact line, as previously measured experimentally, consistently point to a strong Shuttleworth effect. Finally, using Noether’s theorem, we reveal the quantitative link between Young’s law, hysteresis, and the nature of the elastocapillary singularity. Our contribution closes the issue of the missing normal force at the contact line and opens up the development of modern techniques in polymer surface science.

中文翻译：

---

毛细血管Ridge的奇异性质

软固体的机械和化学性质对于生物学和表面科学中的许多应用至关重要。最近的研究使用液滴润湿来探究网状聚合物网络的表面力学，从而引起了有争议的解释。这一争议与长期以来关于液体接触角的杨氏定律悖论有关，后者仅引起水平力平衡。最近的工作表明，对于非常柔软的材料，固体的表面张力在垂直力平衡中起着关键作用，在液滴拉到网状点的确切位置涉及奇异的脊状变形。一个备受争议的问题是，对这种奇异变形的意外测量是否可以归因于非线性体积弹性，或者是否为复杂的表面弹性提供了证据，被称为沙特尔沃思效应。在这里，我们从理论上揭示了弹性高的基底上的弹性毛细管奇异性的性质，具有界面能量的各种本构关系。首先，我们使用目标自适应的有限元模拟精细地解决奇点附近的问题。该模拟证实，体积弹性不会影响接触线处的力平衡。随后，我们导出了非线性弹性的精确解，这些解析性描述了奇点。这些解决方案与数值完全吻合，并且表明，如先前通过实验测量的，接触线处的角度和伸展都始终表明强烈的Shuttleworth效应。最后，使用Noether定理，我们揭示了杨氏定律，磁滞，以及弹性毛细血管奇异性的性质。我们的贡献解决了接触线缺少法向力的问题，并开拓了聚合物表面科学中现代技术的发展。