Diffusion is associated with the random, thermal motion of atoms that produces a change in the macroscopic concentration profile. This process occurs in gases, liquids, amorphous and crystalline solids of metals, ceramics, polymers, semiconductors, etc. The investigation of diffusion provides valuable information about the atomic structure of materials and the defects within them. Perhaps, most importantly, diffusion controls the rates of a wide range of kinetic processes associated with the synthesis of materials, processes by which we modify materials, and processes by which materials fail. The most common driving force for diffusion in a single-phase systems is associated with the entropy of mixing of its constituents (recall that we showed that the entropy of mixing of gases and the components of an ideal solution are always positive—see Sections 1.2.6 and 3.3). Since diffusional processes occur through the thermal motion of atoms (see below), it will not be surprising to learn that the rate of diffusion increases with increasing temperature. However, note that while the mechanisms of thermal motion in gases (random collision of atoms with each other) and liquids (e.g. Brownian motion) necessarily lead to mixing, the mechanisms of mixing within a solid are not as obvious. In solids, thermal motion corresponds to the vibrations of atoms near their equilibrium positions. Since the amplitude of such vibrations is much smaller than the nearest-neighbor separation, it would seem that such thermal motions cannot lead to mixing. Thus, the question ‘‘how do atoms migrate in solids’’ is not so simple. The equations describing diffusion were suggested by the physiologist Fick in 1855 as a generalization of the equations for heat transfer suggested by Fourier in 1824. Fick’s equations for diffusion can be obtained by analogy with Fourier’s equations for heat transfer by replacing heat with the number of atoms, temperature with concentration, and thermal conductivity with diffusivity. Fick’s first law provides a relationship between atomic currents and concentration gradients. As discussed above, this relationship can be understood by analogy with thermal conductivity or electrical conductivity.
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