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硬质颗粒在受限空间中的致密排列具有重要的物理意义，并为许多其他物理系统提供了启发。如何实现硬质颗粒在受限空间中的高密度排列，是一个具有挑战的问题。本文运用蒙特卡洛方法，结合边界压缩机制，研究了二维圆形、正方形和长宽比为5的矩形颗粒在圆形受限空间中的致密排列。具体而言，本文探讨了在压缩过程中不允许颗粒重叠以及允许少数颗粒重叠、后移除重叠（允许临时重叠）两种方法下所能获得的最高密度。研究发现，允许临时重叠的方法能够实现更高的密排构型。进一步地，本文比较了两种压缩方式下获得的构型的径向分布函数和取向序参量，发现两者具有相似的特征，但允许临时重叠的方式在更大区域内显示出有序性。本文研究结果表明，允许颗粒临时重叠可能是提高受限空间中排列密度的有效途径。The dense packing of hard particles in confined spaces is of broad interest in both mathematics and statistical physics. It relates to classical packing problems under geometric constraints, plays a central role in understanding the self-assembly of microscopic particles such as colloids and nanoparticles, and inspires studies across a wide range of physical systems. However, achieving high packing densities under confinement remains challenging due to anisotropic particle shapes, the discontinuous nature of hard-core interactions, and geometric frustration. In this work, we develop a Monte Carlo scheme that combines boundary compression with controlled, temporary particle overlaps. Specifically, we allow a limited number of overlaps during the compression of a circular boundary, which are subsequently removed via standard Monte Carlo relaxation before further compression steps. We apply this strategy to three types of two-dimensional particles—disks, squares, and rectangles with an aspect ratio of 5:1—confined within a circular boundary. As a control, we also perform simulations using a conventional method that strictly prohibits overlaps throughout. The final configurations from both methods exhibit similar structural features. For hard disks, central particles form a triangular lattice, while those near the boundary become more disordered to accommodate the circular geometry. For hard squares, particles at the center organize into a square lattice, whereas those near the boundary form concentric layers. For rectangles, particles in the central region display local smectic-like alignment within clusters that are oriented nearly perpendicular to one another. Near the boundary, some particles align tangentially along the circular edge. Quantitatively, the temporary-overlap strategy consistently yields denser packings across all particle types. Analysis of 10 independent samples shows higher average and maximal packing densities compared to the conventional method. Further analysis of the radial distribution functions and orientational order parameters reveals that, although both methods produce similar structural features, the overlap-allowed method yields a larger central region exhibiting lattice-like or cluster-like ordering. Our findings suggest that allowing temporary particle overlaps is an effective strategy for generating dense configurations of hard particles under confinement. This approach may be extended to more complex systems, including three-dimensional particles or mixtures with different shapes confined within restricted geometries.
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Keywords:
- hard particles /
- overlap /
- packing density /
- configuration
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