AI 中文总结
本文通过匹配渐近展开法分析小刚性球被Helfrich膜部分吞噬的边界层，推导得到结合阈值等关键结果，其自能规律经非线性形状方程验证。
AI 中文摘要
我们研究流体Helfrich膜对小刚性球的轴对称部分吞噬过程。当颗粒与膜的尺寸比ε很小时，连接包裹帽与周围膜的颈部是弹性边界层，我们通过匹配渐近展开法对其进行分析。主导内表面是悬链面（一种无Helfrich能量的极小曲面），因此部分吞噬的能量由一阶修正项携带，仅出现在ε²ln(1/ε)阶。我们通过求解悬链面的非齐次雅可比方程得到该修正项，该方程受自发曲率和颈部必须匹配的环境平均曲率驱动，且给出了其系数的闭合形式。随后可对边界层进行积分，将颈部替换为极点处的标量自能，使外场可独立于内场闭合。对于耦合张力储库的膜，我们推导得到结合阈值，其结果与张力和自发曲率均无关，还得到了完全包裹阈值及包裹转变的滞后现象。颈部自能规律与完整非线性形状方程的结果一致。
英文摘要
We study the axisymmetric partial engulfment of a small rigid sphere by a fluid Helfrich membrane. When the size ratio $\eps$ between the particle and the membrane is small, the neck that joins the wrapped cap to the surrounding membrane is an elastic boundary layer, and we analyse it by matched asymptotic expansions. The leading inner surface is a catenoid, a minimal surface that stores no Helfrich energy, so that the energy of partial engulfment is carried by the first correction and appears only at order $\eps^{2}\ln(1/\eps)$. We obtain it by solving the inhomogeneous Jacobi equation of the catenoid, forced by the spontaneous curvature and by the ambient mean curvature that the neck must match, and we give its coefficient in closed form. The boundary layer can then be integrated out, and the neck replaced by a scalar self-energy carried at the pole, so that the outer field can be closed independently of the inner one. For a membrane coupled to a tension reservoir we derive the binding threshold, which turns out to be independent of both the tension and the spontaneous curvature, the complete-wrapping threshold, and the hysteresis of the envelopment transition. The neck self-energy law is confirmed against the full nonlinear shape equations.
