Department of Applied Mathematics and Systems, Universidad Autónoma Metropolitana-Cuajimalpa, Artificios 40, México D. F. 01120, Mexico.
J Chem Phys. 2013 Dec 7;139(21):214115. doi: 10.1063/1.4836617.
In this work, we derive a general effective diffusion coefficient to describe the two-dimensional (2D) diffusion in a narrow and smoothly asymmetric channel of varying width, embedded on a curved surface, in the simple diffusion of non-interacting, point-like particles under no external field. To this end, we extend the generalization of the Kalinay-Percus' projection method [J. Chem. Phys. 122, 204701 (2005); Phys. Rev. E 74, 041203 (2006)] for the asymmetric channels introduced in [L. Dagdug and I. Pineda, J. Chem. Phys. 137, 024107 (2012)], to project the anisotropic two-dimensional diffusion equation on a curved manifold, into an effective one-dimensional generalized Fick-Jacobs equation that is modified according to the curvature of the surface. For such purpose we construct the whole expansion, writing the marginal concentration as a perturbation series. The lowest order in the perturbation parameter, which corresponds to the Fick-Jacobs equation, contains an additional term that accounts for the curvature of the surface. We explicitly obtain the first-order correction for the invariant effective concentration, which is defined as the correct marginal concentration in one variable, and we obtain the first approximation to the effective diffusion coefficient analogous to Bradley's coefficient [Phys. Rev. E 80, 061142 (2009)] as a function of the metric elements of the surface. In a straightforward manner, we study the perturbation series up to the nth order, and derive the full effective diffusion coefficient for two-dimensional diffusion in a narrow asymmetric channel, with modifications according to the metric terms. This expression is given as D(ξ)=D(0)/w'(ξ)√(g(1)/g(2)){arctan[√(g(2)/g(1))(y(0)'(ξ)+w'(ξ)/2)]-arctan[√(g(2)/g(1))(y(0)'(ξ)-w'(ξ)/2)]}, which is the main result of our work. Finally, we present two examples of symmetric surfaces, namely, the sphere and the cylinder, and we study certain specific channel configurations on these surfaces.
在这项工作中，我们推导出一个通用的有效扩散系数来描述二维（2D）扩散在一个狭窄而平滑不对称的通道，嵌入在一个弯曲的表面上，在没有外部场的情况下，非相互作用的点状粒子的简单扩散。为此，我们扩展了 Kalinay-Percus'投影方法的推广[J. Chem. Phys. 122, 204701（2005）；Phys. Rev. E 74, 041203（2006）]用于在[L. Dagdug 和 I. Pineda，J. Chem. Phys. 137, 024107（2012）]中引入的不对称通道，将各向异性二维扩散方程投影到弯曲流形上，变成一个根据表面曲率修正的有效一维广义 Fick-Jacobs 方程。为此，我们构造了整个展开式，将边值浓度表示为一个微扰级数。微扰参数的最低阶对应于 Fick-Jacobs 方程，包含一个额外的项，用于表示表面的曲率。我们明确地得到了不变有效浓度的一阶修正，这是一个变量的正确边值浓度，并且我们得到了类似于 Bradley 系数[Phys. Rev. E 80, 061142（2009）]的有效扩散系数的一阶近似，作为表面度量元素的函数。我们以直接的方式研究了直到第 n 阶的微扰级数，并推导了狭窄不对称通道中二维扩散的完整有效扩散系数，根据度量项进行修正。这个表达式为 D(ξ)=D(0)/w'(ξ)√(g(1)/g(2)){arctan[√(g(2)/g(1))(y(0)'(ξ)+w'(ξ)/2)]-arctan[√(g(2)/g(1))(y(0)'(ξ)-w'(ξ)/2)]}，这是我们工作的主要结果。最后，我们给出了两个对称曲面的例子，即球体和圆柱体，并研究了这些曲面上的某些特定通道配置。
