Deuterium transport and trapping in aluminum alloys
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I.
INTRODUCTION
transport of hydrogen in metals during fatigue crack growth has been a subject of considerable interest./-~3 Studies on hydrogen transport and trapping in aluminum alloys during fatigue crack growth have indicated that hydrogen penetrates the fracture surface and gets trapped at trapping sites./4'15 Several studies have been carried out to model this type of behavior. 16'17'18 McNabb and Foster 16 studied the permeation of interstitials through the solid and assumed a uniform distribution of traps within the metal. Caskey and Pillinger 17used a finite difference approach to solve the McNabb and Foster differential equations. Frank et al is pointed out that the assumption of a uniform distribution of traps is not realistic and has limited applicability; they, therefore, modified Caskey and Pillinger's work by considering a nonuniform distribution of trapping sites. However, application of this model to the problem of evolution of densities far from the steady state condition presents several difficulties. The approximations used in getting the solution of the diffusion differential equations are inappropriate for the evolution problems. (These approximations are discussed in detail in Appendix A.) Rectification of those discrepancies makes the diffusion differential equations nonlinear and of second order. The purpose of this paper is to present a simpler model, involving first order diffusion differential equations to describe the evolution of the density of deuterium near surfaces during fatigue crack growth. Various aspects of the model are discussed, and the computer simulation is compared to data from secondary ion mass spectroscopy (SIMS) experiments. It is found that this simple model agrees qualitatively with the experimental results. THE
II.
COMPUTER MODELS
A. Model without Hydrogen Traps For a model without traps, consider the one-dimensional problem of atoms jumping between discrete cells of equal volume with a probability p. That is, p is a probability that if the particle is in cell i, it will jump into cell i - 1 or A.K. ZUREK is with Materials Physics, Rockwell International Science Center, 1049 Camino Dos Rios, Thousand Oaks, CA 91360. R. WALSER, Professor, Department of Electrical Engineering, and H. L. MARCUS, Harry L. Kent Professor of Mechanical Engineering and Materials Science, are both with the University of Texas, Austin, TX 78712. Manuscript submitted March 30, 1982.
METALLURGICALTRANSACTIONS A
cell i + 1 in a time interval At. The change of concentration (p) in time will be equal to At-~t = PPi+I + PPi-t -- 2ppi
[I]
which represents the concentration change in cell i, by allowing the particle to jump into cell i from cells i + 1 and i - 1 and allowing cell i to release the concentration to any adjacent cell. It is necessary to determine the jump probability for the first two cells in the presence of a surface that is partially permeable. Cell i = 0 is defined in Figure 1 as being outside the surface of the material. Cell i - 1 is defined as being on the surface itse
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