Meaning
Mathematical method of transport analysis resolves the concentration-dependent diffusion coefficient of a binary solid-state system from a single measured concentration profile. The boltzmann matano transformation simplifies the partial differential equation of Fick’s second law into an ordinary differential equation by introducing a composite variable representing the ratio of position to the square root of time. This approach requires a planar starting interface and sufficient diffusion time to establish a measurable concentration gradient without the boundaries of the diffusion couple affecting the profile.
The method finds extensive use in determining interdiffusion coefficients in metal alloys, semiconductors, and multi-component oxide systems where diffusion rates vary with local composition. It relies on locating the matano interface, which is the plane where the net transport of the two diffusing species is equal and opposite. This reference plane must be established to calculate the integrals and gradients needed for the analysis.
The technique applies to systems with a constant density and fails when there is a significant volume change during diffusion.
Mathematical Principle
Differential equations governed by concentration-dependent diffusion are resolved by substituting a single variable that combines distance and time. This mathematical substitution allows Fick’s second law, which is a partial differential equation, to be integrated directly. The integration of the concentration profile yields the diffusion coefficient as a function of concentration.
This is accomplished by calculating the area under the concentration-distance curve on both sides of the matano interface. The slope of the concentration profile at any given point is also determined, and the diffusion coefficient is calculated from the ratio of the area to this slope. This procedure is performed across the entire concentration range to map the dependency of diffusion on alloy composition.
Operational Execution
Industrial research laboratories employ this method to analyze diffusion couples after high-temperature thermal treatments. The experimental setup involves joining two polished blocks of different compositions and annealing them in a vacuum furnace. After cooling, the couple is sectioned perpendicular to the original interface, and the composition profile is measured using electron probe microanalysis or energy-dispersive spectroscopy.
These concentration data are then plotted against distance to generate the experimental profile. A computer program locates the matano interface by balancing the areas on either side of the interface, ensuring that the mass balance of the diffusing species is preserved. The derivative and integral at each composition point are then computed to extract the diffusion coefficient.
Material Constraint
Accurate application of this analytical method is restricted to binary systems with negligible molar volume change across the diffusion zone. If the molar volume of the alloys varies significantly with composition, the standard analysis introduces large errors. In such cases, the Sauer-Freise method, which is a modified version of the technique, must be used to account for volume changes.
Another major constraint is the requirement that the diffusion zone must not reach the physical ends of the specimen. If the concentration profile is truncated by the boundaries of the sample, the integration cannot be performed, and the method becomes invalid. Additionally, the presence of grain boundaries, dislocations, or other high-diffusivity paths will distort the concentration profile, leading to inaccurate results that do not represent true bulk interdiffusion.
