关键词: 一维半无限扩散, 菲克第二定律, 玻尔兹曼变量代换法, 拉普拉斯变换法

Abstract: One-dimensional semi-infinite solid diffusion serves as a key theoretical basis for engineering processes such as material surface strengthening (e.g., carburizing/nitriding) and semiconductor doping. Its analytical solution based on Ficks Second Law is crucial for grasping diffusion laws. However, there exist teaching pain points in the treatment of this analytical solution in current basic materials science textbooks: the solution is generally derived from the one-dimensional infinite diffusion couple model (by welding two metal rods of equal diameter into a diffusion couple under pressure, solving Ficks second law, and then regarding semi-infinite diffusion as a special case). This approach, nevertheless, tends to cause confusion among students. Specifically, for scenarios where the surface concentration is zero, such as vacuum degassing and surface decarburization of solid plates, tracing back to the infinite model requires assuming a "negative concentration" metal rod, whose physical meaning is difficult to explain and instead increases the complexity of teaching. This paper conducts research to address this issue: first, it clarifies the irrationality of deriving the semi-infinite diffusion formula from the infinite diffusion couple and the root cause of students confusion; subsequently, it proposes a direct derivation approachby combining the initial and boundary conditions of one-dimensional semi-infinite diffusion, solving Ficks Second Law through the Boltzmann variable substitution method, and obtaining an unambiguous concentration distribution formula. Meanwhile, taking carburizing as an example, it introduces the concise solution process of the Laplace transform method to verify the consistency of the results. In addition, this paper answers students questions regarding the concavity and convexity of diffusion curves, pointing out that the concavity and convexity of the curve are determined by the sign of 2C/x2 (the curve is concave downward when the concentration increases and convex upward when the concentration decreases). This is consistent with the inherent nature of diffusion tending to achieve uniform concentration and cannot be altered. This research helps students gain an in-depth understanding of the physical connotation of semi-infinite diffusion and provides support for mastering the core knowledge of the diffusion chapter.

Key words: one-dimensional semi-infinite solid diffusion, Ficks second law, Boltzmann variable substitution method, Laplace transform method