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. Various studies show that these reactions lead to a modification of the model, including also non linear terms. The validity of the model is carefully checked with several tests in 1D, 2D and different geometries.
翻译:在本文中,我们提出并验证了一个用于描述在陷阱边界存在的情况下粒子扩散的多尺度模型。 我们从漂移-扩散方程式开始,其中漂移术语描述泡沫陷阱的影响,并以短范围潜力为模型,具有吸引力和令人厌恶的核心。 泡泡表面吸引的微粒的相互作用由Lennard-Jones 模拟,该颗粒由于离子的疏水特性而简化了捕获过程。 在我们的模型中,这种潜力的效果由大规模保护和无线分析得出的适当边界条件所取代。 假定这种潜力的范围小于$\varepsilon$。 考虑用$\varepsilon 的微量扩张,通过在膨胀中保留最低的顺序条件获得边界条件。 我们调查的另一个方面是泡沫表面附近高浓度产生的饱和效应。 各种研究表明,这些反应导致模型的修改, 包括非线性条件。 模型的有效性在1D、 2D 和不同的地貌中经过若干次的仔细检查。