In metallurgy, the Darken equations are used to describe the solid-state diffusion of materials in binary solutions.
[1] The equations apply to cases where a solid solution's two components do not have the same coefficient of diffusion.
Darken's second equation is: where: In deriving the first equation, Darken referenced Simgelskas and Kirkendall's experiment, which tested the mechanisms and rates of diffusion and gave rise to the concept now known as the Kirkendall effect.
[2] For the experiment, inert molybdenum wires were placed at the interface between copper and brass components, and the motion of the markers was monitored.
The experiment supported the concept that a concentration gradient in a binary alloy would result in the different components having different velocities in the solid solution.
In establishing the coordinate axes to evaluate the derivation, Darken refers back to Smigelskas and Kirkendall’s experiment which the inert wires were designated as the origin.
[1] In respect to the derivation of the second equation, Darken referenced W. A. Johnson’s experiment on a gold–silver system, which was performed to determine the chemical diffusivity.
In this experiment radioactive gold and silver isotopes were used to measure the diffusivity of gold and silver, because it was assumed that the radioactive isotopes have relatively the same mobility as the non-radioactive elements.
If the gold–silver solution is assumed to behave ideally, it would be expected the diffusivities would also be equivalent.
[1] This finding led Darken to analyze Johnson's experiment and derive the equation for chemical diffusivity of binary solutions.
As stated previously, Darken's first equation allows the calculation of the marker velocity
For this equation to be applicable, the analyzed system must have a constant concentration and can be modeled by the Boltzmann–Matano solution.
For the derivation, a hypothetical case is considered where two homogeneous binary alloy rods of two different compositions are in contact.
Here, inert markers are defined to be a group of particles that are of a different elemental make-up from either of the diffusing components and move in the same fashion.
For this derivation, the inert markers are assumed to be following the motion of the crystal lattice.
This results in the total rate of transport for the system being influenced by both factors, diffusion and advection.
[1] The derivation starts with Fick's first law using a uniform distance axis y as the coordinate system and having the origin fixed to the location of the markers.
It is assumed that the markers move relative to the diffusion of one component and into one of the two initial rods, as was chosen in Kirkendall's experiment.
The coordinate system is transformed using a Galilean transformation, y = x − νt, where x is the new coordinate system that is fixed to the ends of the two rods, ν is the marker velocity measured with respect to the x axis.
This transformation then yields The above equation, in terms of the variable x, only takes into account diffusion, so the term for the motion of the markers must also be included, since the frame of reference is no longer moving with the marker particles.
This result is similar to Fick's second law, but with an additional advection term: The same equation can be written for the other component, designated as component two: Using the assumption that C, the total concentration, is constant,[3] C1 and C2 can be related in the following expression: The above equation can then be used to combine the expressions for
At relative infinite distances from the initial interface, the concentration gradients of each of the components and the marker velocity can be assumed to be equal to zero.
[1] This derivation was the approach taken by Darken in his original 1948, though shorter methods can be used to attain the same result.
To derive Darken's second equation the gradient in Gibb's chemical potential is analyzed.
[1] To begin, the flux J is equated to the product of the differential of the gradient and the mobility B, which is defined as the diffusing atom's velocity per unit of applied force.
One application in which Darken’s equations play an instrumental role is in analyzing the process of diffusion bonding.
[6] Diffusion bonding is used widely in manufacturing to connect two materials without using adhesives or welding techniques.
[6] In a similar manner, Darken’s equations were used in a paper by Watanabe et al., on the nickel-aluminum system, to verify the interdiffusion coefficients that were calculated for nickel aluminum alloys.
[7] Application of Darken’s first equation has important implications for analyzing the structural integrity of materials.
[8] Use of Darken’s equation in this form has important implications for determining the flux of vacancies into a material undergoing diffusion bonding, which, due to the Kirkendall effect, could lead to porosity in the material and have an adverse effect on its strength.