In this paper we propose and validate a multiscale model for the description of particle diffusion in presence of trapping boundaries. We start from a drift-diffusion equation in which the drift term describes the effect of bubble traps, and is modeled by a short range potential with an attractive term and a repulsive core. The interaction of the particles attracted by the bubble surface is simulated by the Lennard-Jones potential that simplifies the capture due to the hydrophobic properties of the ions. In our model the effect of the potential is replaced by a suitable boundary condition derived by mass conservation and asymptotic analysis. The potential is assumed to have a range of small size $\varepsilon$. An asymptotic expansion in the $\varepsilon$ is considered, and the boundary conditions are obtained by retaining the lowest order terms in the expansion. Another aspect we investigate is saturation effect coming from high concentrations in the proximity of the bubble surface. The validity of the model is carefully checked with several tests in 1D, 2D and different geometries.

翻译：本文提出并验证了一种用于描述存在俘获边界时颗粒扩散的多尺度模型。我们从漂移-扩散方程出发，其中漂移项描述了气泡俘获效应，并通过具有吸引势和排斥核的短程势能进行建模。受气泡表面吸引的颗粒间相互作用采用伦纳德-琼斯势进行模拟，该势能简化了因离子疏水特性导致的捕获过程。在模型中，势能效应由基于质量守恒和渐近分析推导出的合适边界条件替代。假设势能的作用范围尺度较小，记为$\varepsilon$。通过考虑$\varepsilon$的渐近展开，保留展开式中的最低阶项即可获得边界条件。我们研究的另一个方面是气泡表面附近高浓度引发的饱和效应。通过一维、二维及不同几何构型中的多项测试，模型的可靠性得到了充分验证。