Motivated by various physical, cellular and ecological applications, there has been a recent resurgence of interest in studying the boundary adsorption-desorption of diffusive substances between a bulk (body) and a surface, by using “bulk-surface models” (involving volumetric densities and surface densities) or by using models involving dynamical boundary conditions for volumetric densities. The surface in the models has no thickness. However, in some real world applications, the “surface” is actually a thin membrane, having positive, albeit small thickness δ, as in the cases of thermal barrier coatings for turbine engine blades and cell membranes. In such situations, rigorous derivations for these models seem lacking and desirable. In this paper, we start with two full models each of which contains reaction-diffusion equations in the bulk and the thin membrane, respectively, with two types of reasonable transmission conditions linking the two. Then in the limit of $\delta \to 0$, we obtain two effective models, with one being a bulk-surface model and the other being the dynamical boundary value problem model, from which we can also recover the surface density of the other substance. Our analysis reveals that to have such effective models, it is crucial that (i) appropriate transmission conditions are satisfied on the interface between the bulk and the thin membrane, and (ii) the diffusion tensor in the thin membrane is “optimally aligned”, which means the principal axes of the tensor are either normal or tangential to the interface.

受各种物理，细胞和生态学应用的启发，最近出现了一种兴趣浓厚的兴趣，即通过使用“体表模型”（涉及体积密度）来研究散装物质（物体）与表面之间的扩散性物质的边界吸附-解吸。和表面密度）或使用涉及体积密度动态边界条件的模型。模型中的表面没有厚度。但是，在某些实际应用中，“表面”实际上是薄膜，尽管厚度δ小，但具有正值，例如用于涡轮发动机叶片和细胞膜的隔热涂层。在这种情况下，对这些模型的严格推导似乎是缺乏和可取的。在本文中，我们从两个完整的模型开始，每个模型分别包含在本体和薄膜中的反应扩散方程，并通过两种合理的传递条件将两者联系在一起。然后在$\delta \to 0$，我们获得了两个有效模型，一个是体表面模型，另一个是动力学边界值问题模型，从中我们还可以恢复其他物质的表面密度。我们的分析表明，要拥有这样一种有效的模型，至关重要的是：（i）在本体与薄膜之间的界面上满足适当的传输条件，并且（ii）薄膜中的扩散张量“最佳对齐”，这意味着张量的主轴与界面正交或相切。