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the mass of material reacted per unit area for a unit time. These can easily be converted to the depth of penetration per unit time by dividing by the density of the material as shown below:

(2.71)

where:
P	=	depth of penetration
M	=	mass loss

ρ= density

A	=	area of exposure
t	=	time of exposure

In using the above equation to calculate corrosion rates from laboratory experiments, one must be very conscious of the total surface area exposed to corrosion. This will include a determination of the open porosity of the specimen. Many investigators have attempted to compare corrosion resistance of various materials incorrectly by omitting the porosity of their samples. Omitting the porosity, although not giving a true representation of the material’s corrosion, will give a reasonable idea of the corrosion of the as-manufactured material.

2.9 DIFFUSION

When the transport of ions or molecules occurs in the absence of bulk flow, it is called diffusion. Substances will spontaneously diffuse toward the region of lower chemical potential. This transport or flux of matter is represented by Fick’s first law and is proportional to the concentration gradient. This is represented by:

(2.72)

where:
Jix	= flux of component i in the x direction
D	=	diffusion coefficient
ci	=	concentration of component i

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The How of material is thus proportional to the concentration gradient and is directed from the region of high concentration to one of low concentration.

Fick’s second law describes the nonstationary state of flow where the concentration of a fixed region varies with time:

(2.73)

Since diffusion is directional, one must be aware of anisotropic effects. The rate of diffusion may be very different in different crystalline directions. In isometric crystals, the diffusion coefficient is isotropic, as it is in polycrystalline materials as long as no preferred orientation exists. The second-order tensor defined by the equations for the flux, J, in each of the x, y, and z directions, contains a set of nine diffusion coefficients designated Dij. Due to the effects of the various symmetry operations in the tetragonal, hexagonal, orthorhombic, and cubic crystal classes, only a few of these Dij have nonzero values. All the off-diagonal Dij (i=j) are equal to zero. Thus only the three diagonal values are of any consequence; however, symmetry again causes some of these to be equivalent. In the remaining two crystal classes, the number of independent coefficients increases; however, the total number is decreased somewhat since Dij=Dji. The possible nonzero diffusion coefficients for each of the crystal classes are shown in Table 2.9.

A solution of Eq. (2.73) for nonsteady-state diffusion in a semi-infinite medium (D is independent of concentration) is:

(2.74)

where:
C(x,t) =		concentration after time t
Co	=	initial concentration in the medium

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TABLE 2.9 Effect of Symmetry Upon the Second-Rank Tensor Diffusion Coefficients

Solutions to Eq. (2.74) depend upon the boundary conditions that one selects in the evaluation. More than one set of boundary conditions have been selected by various investigators, and thus several solutions to the equation exist in the literature that may provide some confusion to the uninitiated. In the above case [Eq. (2.74)], which is appropriate for the diffusion between two solids, the boundary conditions were selected such that as time passes, the diffusing species are depleted on one side of the boundary and increased on the other. This will yield a constant midpoint concentration at the boundary of Co/2. In the case of corrosion of a solid by a liquid, one assumes that the concentration of diffusing species from the liquid into the solid remains constant at the boundary (Cs) at a value equal to that in the bulk. The solution to Eq. (2.73) is then:

(2.75)

where Cs is the concentration at the surface. One should note that the sign within the brackets changes when the boundary conditions are changed. is a measure of the order of magnitude of the distance that an average atom will travel and

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thus approximates the distance over which the concentration will change during diffusion. The use of error functions (erf) in evaluating diffusion is relatively easy by use of published tables [2.143] for various values of erf(z).

Most of the solutions to Fick’s equations assume that D is constant; however, in most real cases, the diffusion coefficient can vary with time, temperature [see Eq. (2.40)], composition, or position along the sample, or any combination of these. If these are included in the equation, the mathematics become very difficult if not impossible; thus the equations used to describe diffusion generally assume constant D. See Table 2.10 for some typical values of diffusion coefficients.

TABLE 2.10 Diffusion Coefficients for Some Typical Ceramics

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Several mechanisms for diffusion have been hypothesized and investigated. One of the more important in ceramic materials is diffusion by vacancy movement in nonstoichiometric materials. Another mechanism involves diffusion by movement from one interstitial site to another. The ease with which this mechanism can occur, however, is not as great as that by vacancy movement. Other mechanisms that provide high-diffusivity paths include diffusion aided by dislocations, free surfaces, or grain boundaries.

Permeability constants as a function of temperature give an indication of the ease of diffusion of a species through a material. Silica has the lowest permeability to oxygen. This has been attributed to the difference in mechanism of transport among silica and most other materials. Transport in silica is by molecular species, whereas in other materials, it is by ionic species [2.144]. For this reason, silica-forming reactions are the most desirable for protection against oxygen diffusion

[2.145].

Since many applications of ceramics involve thermal gradients, some mention of thermal diffusion should be made. Based upon studies in liquids, this has been called the Sorét effect. To evaluate the effect using Fick’s first law, an additional term must be added to Eq. (2.72) that involves the temperature gradient. The flux is then given by:

(2.76)

where ßi is a constant independent of the thermal gradient for component i and may be positive or negative depending upon whether diffusion is down or up the thermal gradient, respectively. This constant is proportional to D and is given by:

(2.77)

where Qi* is an empirical parameter that describes the sign and magnitude of the thermal diffusion effect. It has also been called the heat of transport. One interesting phenomenon that

Copyright © 2004 by Marcel Dekker, Inc.