Fig. 1 Schematic illustration of solute segregation along the secondary dendrite arm spacing (SDAS).

A homogeneous chemical composition and microstructure always is desired and seldom attained in practice. Corrosion resistance, age hardening and machinability are just a few material characteristics that are affected by chemical segregation. Chemical segregation creates regions that are either anodic or cathodic relative to the average composition. In a corrosive environment, these regions function as tiny batteries and the anodic regions corrode. Segregation also affects heat treatment of aluminum and steel. In the case of an 356 aluminum alloy casting, solute-rich regions overage, while underaging occurs in the solute-lean regions. On average, the casting may be of the correct hardness, but the microstructure is far from ideal or expected. Homogenization heat treatments often are used to reduce the degree of segregation and improve performance.

The mathematics of diffusion is relatively simple in one dimension; however, the complexity increases dramatically when real microstructures are considered. For example, Fig. 1 shows a hypothetical solute concentration profile measured along the length of a dendrite and intersecting the secondary dendrite arms. As a one-dimensional (1-D) problem, the solute profile can be handled by a series of cosine or sine terms. A more realistic approach would be to consider the secondary arms as an array of cylinders and consider diffusion in a radial direction. Solution of the latter problem requires a series of Bessel functions. A comparison of the 1-D and 2-D solutions shows that the 1-D case always requires a longer time to achieve the same degree of homogenization. Thus, the simpler 1-D case represents an upper bound to the required time.

Calculation of homogenization time usually is complicated by the uncertainty in the chemical diffusivity. It often is sufficient to assume that the slowest diffusing species controls chemical diffusivity. However, even this assumption can lead to significant errors. Table I shows the results (available in the literature) of Chromium (Cr) diffusion in various iron-base alloys. Note that the three orders of magnitude difference of the diffusivities at 2190F (1200C) should be of more concern than errors associated with using a 1-D model to calculate the time required for homogenization.

Fig. 2 Schematic illustration showing the definition of the segregation index, d, and how the solute profile changes during homogenization.

During homogenization, the solute profile is leveled out to an average composition (Co) as shown in Fig. 2.

Fig. 3 Schematic illustration showing how the segregation index, d_ changes with time and temperature. The temperature dependence is contained in the diffusivity term D.
If the exact chemical profile is not required, the diffusion problem can be reduced to examining the segregation index, d, which is a measure of the maximum solute amplitude relative to the composition average. Figure 3 shows that d decreases exponentially with time and temperature.

A 1-D solution for the segregation index is given by the following equations where SDAS is defined as the secondary dendrite arm spacing,tis the time,Tis the absolute temperature andRis the universal gas constant. This solution is very convenient because a homogenization heat treatment can be formulated without any knowledge of the actual chemical segregation other than the SDAS. If an 80% reduction in the segregation index is required, d is set to 0.2 and the time to reach this index is calculated for the appropriate temperature.