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12.diffusion - T Y Tan 12 DIFFUSION At a non-zero...

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T. Y. Tan 1 12. DIFFUSION At a non-zero temperature, atomic motion occurs in all materials in all states. For example, thermal energy keeps the atoms of a gas constantly in random motion. If the motion of atoms (or any other particles, e.g., electrons in a semiconductor) is directional, it leads to a directional mass flow or mass transport. The process producing a directional mass flow by the motion of individ- ual atoms is called diffusion . Directional mass flow may obviously also be achieved by transla- tional motions of a group of atoms collectively, which are not diffusion processes. Diffusion is induced by the presence of a chemical potential gradient inside the material. Here the term chemical potential is defined in the general sense so that effects due to all possible physical or chemical causes are included. Examples of these causes are the entropy of mixing, strain field, electric field, thermal field, etc. Some of these may be applied from outside the material, e.g., an electrical field or a thermal gradient. Diffusion is such an interesting phenomenon that it has at- tracted studies from an intrinsic point of view in all materials. Diffusion in solids is also of great importance from a practical point of view. Phase changes in alloys involve a redistribution of various kinds of atoms, which controls the rate of the phase change. For semiconductors, the dif- fusion process is used to fabricate pn-junctions in electronic devices. For these reasons, extensive studies have been carried out in all the technologically important solid state materials. The species that exhibit a diffusion process include not only solute or impurity atoms, but also the matrix crystal self-atoms (self-diffusion) as well as point defects. 12.1 The Statistic Nature of Diffusion The first way of describing diffusion assumes that the process is due to numerous random motions of the individual atoms, and the result is a statistical average of such motions. Thermal agitation supplies an atom occasionally with enough energy to overcome an energy barrier named the activation energy (e.g., the energy needed to break bonds) to jump from one atomic site to a neighboring one. These energy fluctuations arise from collisions of the atom with its neighbors or due to collisions with many phonons. The atom thus 'thrown off' migrates in an ar- bitrary direction. The migration path of the atom is thus an unpredictable random one. Neverthe- less, for a large number of atoms making such movement they produce a directional flow down the concentration gradient. For a plane perpendicular to the concentration gradient, even though the atoms migrate randomly, more atoms will pass the plane from the high concentration side than from the other, because there are more atoms on one side of the plane than the other.
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T. Y. Tan 2 12.1.1 Fick's Laws To develop the above argument quantitatively, consider the following idealized one dimen- sional case: (i) The volume solute concentration C varies along the x-axis; (ii) The concentration gradient, C/
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