Interaction between diffusion and chemical stresses

Abstract

The present work studies the interaction between chemical stresses and diffusion. A new relation between hydrostatic stress and concentration of solute atoms is established. For a solid free of action of body force, the Laplacian of the hydrostatic stress is proportional to the Laplacian of the concentration of solute atoms, that is, deviation of the hydrostatic stress from its local average is proportional to deviation of the local concentration of solute atoms. A general relationship among surface concentration of solute atoms, normal stress and surface deformation of a solid is then derived, in which the normal stress is dependent on the mean curvature of the undeformed surface and tangential components of the surface displacement. A closed-form solution of the steady state concentration of solute atoms in a thin plate is obtained. It turns out that linear distribution of solute atoms in the plate is non-existent due to the interaction between chemical stresses and diffusion.

Introduction

It is known that diffusion of atoms in materials could lead to the evolution of local stresses, which has been referred as diffusion-induced stresses or chemical stresses [1], [2]. The effect of chemical stresses on material behavior can be observed in different material systems and be used in a variety of microelectronic and micromechanical systems. For examples, the use of Cu and low-K dielectrics in high-performance CMOS devices imposes a challenge in the control of the diffusion of Cu as degradation of time-dependent dielectric breakdown (TDDB) in Cu metallization is related to the diffusion of Cu ion caused by electrical stress and the TDDB lifetime depends on the breakdown voltage of the low-K material. Impurities have been doped in semiconductor materials to alter the Fermi level for the design of microelectronic devices [3], while the diffusion-induced stresses could initiate generation of dislocations and affect distribution of impurities and related electronic behavior. Diffusion of hydrogen into metals can introduce hydrogen-induced cracking and hydrogen-enhanced local plasticity [4]. The evolution of stresses during diffusion was originally analyzed by Prussin [1]. Li [2] studied the diffusion-induced stresses in elastic media of simple geometries. Lee and co-workers [5], [6] analyzed the effect in composite materials. Larch and Cahn [7], [8] investigated the stresses arising from inhomogeneities in materials. Recently, Yang and Li [9] considered the effect of chemical stresses on the bending of beam/plate structure.

The effects of stresses on creep deformation involving diffusion of vacancies were proposed by Nabarro [10] and Herring [11]. Li addressed the effect of stresses on the effective diffusivity for a given stress field [2] and Chu and Li [12] analyzed the stress-induced diffusion in the impression of a half-space. Yang and Li [13], [14] considered the stress-induced diffusion in the impression of thin films. Based on the concept of thermodynamic factor, Alefeld et al. [15] solved the problem of diffusion when subjected to applied stress. Larch and Cahn [7], [8] analyzed the effect of stresses on local diffusion in a solid, where the coupling between diffusion and chemical stresses was addressed. The effect of chemical stresses on diffusion was discussed by Chu and Lee [16] using Li's solution [2], while they did not consider the coupling effect. Kandasamy [17] addressed the influence of chemical stresses on space-time variation of concentration during diffusion of hydrogen in a palladium alloy and proposed a quadratic steady-state hydrogen concentration profile. Using Li's solution [2] and neglecting the coupling effect, Zhang et al. [18] reanalyzed the distribution of the steady-state concentration of hydrogen in elastic membranes during hydrogen diffusion. They claimed that nonlinear distribution of concentration is non-existent and linear form is the only solution for ideal solid solution. Recently, Aziz analyzed the effects of pressure and stress on diffusion from atomic point view [19] and considered such effects on diffusion in Si [20].

Considering the importance of the coupling between diffusion and chemical stresses in the interpretation of stress relaxation and in the understanding of up-hill diffusion effect in hydrogen permeable metals, we consider the coupling effect on the distribution of concentration of solute atoms in a thin plate. Some new relations between chemical stresses and concentration of solute atoms are formulated, from which an analytical solution of the distribution of the steady-state concentration in a plate is obtained.