Simulation of Carbon Diffusion in Steel Driven by a Temperature Gradient

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ection I: Basic and Applied Research

Simulation of Carbon Diffusion in Steel Driven by a Temperature Gradient Lars Ho¨ glund and John A˚ gren

(Submitted November 16, 2009) The basis of thermomigration in multicomponent alloys is summarized, and the general equations are given and implemented in the DICTRA software. Experimental information from Okafor et al. is analyzed with the new simulations and it is concluded that steady-state conditions was not established during their experiment. A heat of transport QC ¼ 44000 J/mol, almost four times larger than the value given by Okafor et al., was found to give a satisfactory representation of the experimental information.

Keywords

heat of transport, mobility, steady state, thermomigration

full equations has been implemented in the DICTRA software, and the experimental information given by Okafor et al.[3] has been analyzed with DICTRA simulations.

1. Introduction

2. Temperature Gradients and Diffusion

A temperature gradient, imposed on a chemically homogeneous material, may cause diffusional transport of atoms, so-called thermomigration, see for example the text book by Philibert.[1] The phenomenon is technically important whenever a component is subjected to a temperature gradient during prolonged times, e.g., in superheaters, heat exchanger tubes, etc., made by steels or Ni-base alloys. The effect is more pronounced for the mobile interstitials than for the sluggish substitutional elements. However, it may also be important even during quite short times if the gradient is sufficiently strong, e.g., in solder joints. Thermomigration stems from the cross effects described by the general phenomenological equations of irreversible thermodynamics, see for example Ref 2. These equations express the fluxes of various quantities as linear functions of gradients of the conjugated thermodynamic potentials. The basic equations become particularly simple if steady-state conditions have been established and it is then straight forward to evaluate a quantity called the heat of transport from a concentration profile once the temperature profile is known. This approach has been used by several authors, see for example Okafor et al.[3] and Mathuni et al.[4] ˚ gren[5] developed a In the early 1990s, Andersson and A general formalism to represent diffusion data for multicomponent systems in terms of intrinsic mobilities. Their method was implemented in the DICTRA software[6] and has now become widely used to analyze diffusion. However, it was not capable of taking temperature gradients into account. In the present report, a numerical solution of the

2.1 General Case and the Heat of Transport

˚ gren, Division of Physical Metallurgy, Lars Ho¨glund and John A Department of Materials Science and Engineering, Royal Institute of Technology, 100 44 Stockholm, Sweden. Contact e-mail: [email protected].

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The phenomenological equations for diffusional flux Jk of species k, in the presence of both a chemical potential and a temperature gradient of an n component system may be written, Jk ¼ 

n