Magnetic quadrupole modules form stable assemblies with arbitrary two-dimensional shapes and magnetizations. Magnetic dipole-dipole interactions govern the behavior of magnetic matter across scales from micrometer colloidal particles to centimeter magnetic soft robots. This pairwise long-range interaction creates rich emergent phenomena under both static and dynamic magnetic fields. However, magnetic dipole particles, from either ferromagnetic or paramagnetic materials, tend to form chain-like structures as low-energy configurations due to dipole symmetry. The repulsion force between two magnetic dipoles raises challenges for creating stable magnetic assemblies with complex two-dimensional (2D) shapes. In this work, we propose a magnetic quadrupole module that is able to form stable and frustration-free magnetic assemblies with arbitrary 2D shapes. The quadrupole structure changes the magnetic particle-particle interaction in terms of both symmetry and strength. Each module has a tunable dipole moment that allows the magnetization of overall assemblies to be programmed at the single module level. We provide a simple combinatorial design method to reach both arbitrary shapes and arbitrary magnetizations concurrently. Last, by combining modules with soft segments, we demonstrate programmable actuation of magnetic metamaterials that could be used in applications for soft robots and electromagnetic metasurfaces.

中文翻译：

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具有任意形状和磁化强度的磁性四极组件

磁性四极模块可形成具有任意二维形状和磁化强度的稳定组件。磁偶极子-偶极子相互作用控制着从微米级胶体粒子到厘米级磁性软机器人的各种尺度上的磁性物质的行为。这种成对的远程相互作用会在静态和动态磁场下产生丰富的出现现象。但是，由于偶极子的对称性，来自铁磁性或顺磁性材料的磁性偶极子粒子倾向于形成链状结构作为低能构型。两个磁偶极子之间的排斥力对创建具有复杂二维（2D）形状的稳定磁性组件提出了挑战。在这项工作中，我们提出了一种磁性四极模块，该模块能够形成具有任意2D形状的稳定且无挫折的磁性组件。四极结构在对称性和强度方面都改变了磁性粒子与粒子之间的相互作用。每个模块都有一个可调的偶极矩，可在单个模块级别对整个组件的磁化进行编程。我们提供了一种简单的组合设计方法，可以同时达到任意形状和任意磁化强度。最后，通过将模块与软段组合，我们演示了可用于软机器人和电磁超表面应用的磁性超材料的可编程致动。我们提供了一种简单的组合设计方法，可以同时达到任意形状和任意磁化强度。最后，通过将模块与软段组合，我们演示了可用于软机器人和电磁超表面应用的磁性超材料的可编程致动。我们提供了一种简单的组合设计方法，可以同时达到任意形状和任意磁化强度。最后，通过将模块与软段组合，我们演示了可用于软机器人和电磁超表面应用的磁性超材料的可编程致动。