In
metallurgy
Metallurgy is a domain of materials science and engineering that studies the physical and chemical behavior of metallic elements, their inter-metallic compounds, and their mixtures, which are known as alloys.
Metallurgy encompasses both the sc ...
, the Darken equations are used to describe the solid-state diffusion of materials in binary solutions. They were first described by
Lawrence Stamper Darken
Lawrence Stamper Darken (18 Sept. 1909, Brooklyn NY – 7 June 1978, State College PA) was a physical chemist and metallurgist, known for his two equations describing solid-state diffusion in binary solutions.
He earned his bachelor's degree in ...
in 1948.
[Darken, L. S. "Diffusion, mobility and their interrelation through free energy in binary metallic systems". Trans. AIME 175.1 (1948): 184–194.] The equations apply to cases where a solid solution's two components do not have the same
coefficient of diffusion.
The equations
Darken's first equation is:
:
where:
*
is the ''marker velocity'' of inert markers showing the diffusive flux.
*
and
are the diffusion coefficients of the two components.
*
and
are the
atomic fractions of the two components.
*
represents the direction in which the diffusion is measured.
It is important to note that this equation only holds in situations where the total concentration remains constant.
Darken's second equation is:
:
where:
*
is the
activity coefficient
In thermodynamics, an activity coefficient is a factor used to account for deviation of a mixture of chemical substances from ideal behaviour. In an ideal mixture, the microscopic interactions between each pair of chemical species are the same (or ...
of the first component.
*
is the overall diffusivity of the binary solution.
Experimental methods
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 The Kirkendall effect is the motion of the interface between two metals that occurs as a consequence of the difference in diffusion rates of the metal atoms. The effect can be observed for example by placing insoluble markers at the interface betwee ...
. 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. The experiment showed that in brass zinc had a faster relative velocity than copper, since the molybdenum wires moved farther into the brass. 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.
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. Therefore, the overall diffusion coefficient of the system would be the average of each components diffusivity; however, this was found not to be true.
This finding led Darken to analyze Johnson's experiment and derive the equation for chemical diffusivity of binary solutions.
Darken's first equation
Background
As stated previously, Darken's first equation allows the calculation of the marker velocity
in respect to a binary system where the two components have different diffusion coefficients. 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. The sides are protected, so that all of the diffusion occurs parallel to the length of the rod. In establishing the coordinate axes to evaluate the derivation, Darken sets the x-axis to be fixed at the far ends of the rods, and the origin at the initial position of the interface between the two rods. In addition this choice of a coordinate system allows the derivation to be simplified, whereas Smigelskas and Kirkendall's coordinate system was considered to be the non-optimal choice for this particular calculation as can be seen in the following section. At the initial planar interface between the rods, it is considered that there are infinitely small inert markers placed in a plane which is perpendicular to the length of the rods. 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
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
: \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
. The motion relative to the marker is associated with
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
,
, while the motion of the markers is associated with
advection
In the field of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is al ...
,
.
Fick’s first law, the previous equation stated for diffusion, describes the entirety of the system for only small distances from the origin, since at large distances advection needs to be accounted for. This results in the total rate of transport for the system being influenced by both factors, diffusion and advection.
Derivation
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. In the following equation, which represents Fick's first law for one of the two components, ''D''
1 is the diffusion coefficient of component one, and ''C''
1 is the concentration of component one:
:
This coordinate system only works for short range from the origin because of the assumption that marker movement is indicative of diffusion alone, which is not true for long distances from the origin as stated before. The coordinate system is transformed using a
Galilean transformation
In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotatio ...
, ''y'' = ''x'' − ν''t'', 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. The variable ''t'', time, is assumed to be constant, so that the partial derivative of ''C''
1 with respect to ''y'' is equal to the partial of ''C''
1 with respect to ''x''. 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. In the equation below,
is the velocity of the markers.
:
Taking the above equation and then equating it to the accumulation rate in a volume results in the following equation. This result is similar to
Fick's second law
Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion eq ...
, 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,
[Sekerka, R. F]
"Similarity Solutions for a Binary Diffusion Couple with Diffusivity and Density Dependent on Composition"
''Progress in Materials Science'' 49 (2004): 511–536. ''C''
1 and ''C''
2 can be related in the following expression:
:
The above equation can then be used to combine the expressions for
and
to yield
:
Since ''C'' is constant, the above equation can be written as
:
The above equation states that
is constant because the derivative of a constant is equal to zero. Therefore, by integrating the above equation it is transforms to
, where
is an integration constant.
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. Based on this condition and the choice for the coordinate axis, where the ''x'' axis fixed at the far ends of the rods, ''I'' is equal zero. These conditions then allow the equation to be rearranged to give
:
Since ''C'' is assumed to be constant,
. Rewriting this equation in terms of atom fraction
and
yields
:
Accompanying derivation
Referring back to the derivation for Darken's first equation,
is written as
:
Inserting this value for
in