Darken's equations

As stated previously, Darken's first equation allows the calculation of the marker velocity ${\displaystyle \nu }$ 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. The motion relative to the marker is associated with diffusion, ${\displaystyle -D_{1}{\tfrac {\partial C_{1}}{\partial y}}}$, while the motion of the markers is associated with advection, ${\displaystyle C_{1}\nu }$. 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.[2]

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, D₁ is the diffusion coefficient of component one, and C₁ is the concentration of component one:

${\displaystyle -D_{1}{\frac {\partial C_{1}}{\partial y}}.}$

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, 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. The variable t, time, is assumed to be constant, so that the partial derivative of C₁ with respect to y is equal to the partial of C₁ with respect to x. This transformation then yields

${\displaystyle -D_{1}{\frac {\partial C_{1}}{\partial x}}.}$

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, ${\displaystyle \nu }$ is the velocity of the markers.

${\displaystyle -\left[D_{1}{\frac {\partial C_{1}}{\partial x}}-C_{1}\nu \right].}$

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, but with an additional advection term:

${\displaystyle {\frac {\partial C_{1}}{\partial t}}={\frac {\partial }{\partial x}}\left[D_{1}{\frac {\partial C_{1}}{\partial x}}-C_{1}\nu \right].}$

The same equation can be written for the other component, designated as component two:

${\displaystyle {\frac {\partial C_{2}}{\partial t}}={\frac {\partial }{\partial x}}\left[D_{2}{\frac {\partial C_{2}}{\partial x}}-C_{2}\nu \right].}$

Using the assumption that C, the total concentration, is constant, C₁ and C₂ can be related in the following expression:

${\displaystyle C=C_{1}+C_{2}.}$

The above equation can then be used to combine the expressions for ${\displaystyle {\tfrac {\partial C_{1}}{\partial t}}}$ and ${\displaystyle {\tfrac {\partial C_{2}}{\partial t}}}$ to yield

${\displaystyle {\frac {\partial C}{\partial t}}={\frac {\partial }{\partial x}}\left[D_{1}{\frac {\partial C_{1}}{\partial x}}+D_{2}{\frac {\partial C_{2}}{\partial x}}-C\nu \right].}$

Since C is constant, the above equation can be written as

${\displaystyle 0={\frac {\partial }{\partial x}}\left[D_{1}{\frac {\partial C_{1}}{\partial x}}+D_{2}{\frac {\partial C_{2}}{\partial x}}-C\nu \right].}$

The above equation states that ${\displaystyle \textstyle D_{1}{\frac {\partial C_{1}}{\partial x}}+D_{2}{\frac {\partial C_{2}}{\partial x}}-C\nu }$ is constant because the derivative of a constant is equal to zero. Therefore, by integrating the above equation it is transforms to ${\displaystyle \textstyle D_{1}{\frac {\partial C_{1}}{\partial x}}+D_{2}{\frac {\partial C_{2}}{\partial x}}-C\nu =I}$, where ${\displaystyle I}$ 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.[5] These conditions then allow the equation to be rearranged to give

${\displaystyle \nu ={\frac {1}{C}}\left[D_{1}{\frac {\partial C_{1}}{\partial x}}+D_{2}{\frac {\partial C_{2}}{\partial x}}\right].}$

Since C is assumed to be constant, ${\displaystyle \textstyle {\frac {\partial C_{1}}{\partial x}}=-{\frac {\partial C_{2}}{\partial x}}}$. Rewriting this equation in terms of atom fraction ${\displaystyle N_{1}={\tfrac {C_{1}}{C}}}$ and ${\displaystyle N_{2}={\tfrac {C_{2}}{C}}}$ yields[2]

${\displaystyle \nu =(D_{1}-D_{2}){\frac {\partial N_{1}}{\partial x}}=(D_{2}-D_{1}){\frac {\partial N_{2}}{\partial x}}.}$

Referring back to the derivation for Darken's first equation, ${\displaystyle \nu }$ is written as

${\displaystyle \nu ={\frac {1}{C}}\left[D_{1}{\frac {\partial C_{1}}{\partial x}}+D_{2}{\frac {\partial C_{2}}{\partial x}}\right].}$

Inserting this value for ${\displaystyle \nu }$ in ${\displaystyle \textstyle {\frac {\partial C}{\partial t}}={\frac {\partial }{\partial x}}\left[D_{1}{\frac {\partial C_{1}}{\partial x}}-C_{1}\nu \right]}$ gives

${\displaystyle {\frac {\partial C_{1}}{\partial t}}={\frac {\partial }{\partial x}}\left[D_{1}{\frac {\partial C_{1}}{\partial x}}-{\frac {C_{1}}{C}}\left[D_{1}{\frac {\partial C_{1}}{\partial x}}+D_{2}{\frac {\partial C_{2}}{\partial x}}\right]\right].}$

As stated before, ${\displaystyle \textstyle {\frac {\partial C_{1}}{\partial x}}=-{\frac {\partial C_{2}}{\partial x}}}$, which gives

${\displaystyle {\frac {\partial C_{1}}{\partial t}}={\frac {\partial }{\partial x}}\left[{\frac {C_{1}+C_{2}}{C}}D_{1}{\frac {\partial C_{1}}{\partial x}}-{\frac {C_{1}}{C}}\left[D_{1}{\frac {\partial C_{1}}{\partial x}}-D_{2}{\frac {\partial C_{1}}{\partial x}}\right]\right].}$

Rewriting this equation in terms of atom fraction ${\displaystyle N_{1}={\tfrac {C_{1}}{C}}}$ and ${\displaystyle N_{2}={\tfrac {C_{2}}{C}}}$ yields

${\displaystyle {\frac {\partial N_{1}}{\partial t}}={\frac {\partial }{\partial x}}\left[(N_{2}D_{1}+N_{1}D_{2}){\frac {\partial N_{1}}{\partial x}}\right].}$

By using ${\displaystyle \lambda \equiv {\tfrac {x}{t^{1/2}}}}$ and solving to the form ${\displaystyle N_{1}=f(\lambda )}$, it is found that

${\displaystyle -{\frac {1}{2}}\lambda \,dN_{1}=d[(N_{2}D_{1}+N_{1}D_{2}){\frac {dN_{1}}{d\lambda }}].}$

Integrating the above gives the final equation:

${\displaystyle D=D_{1}N_{2}+D_{2}N_{1}.}$

This equation is only applicable for binary systems that follow the equations of state and the Gibbs–Duhem equation. This equation, as well as Darken's first law, ${\displaystyle \nu =(D_{2}-D_{1}){\tfrac {\partial N_{2}}{\partial x}}}$, gives a complete description of an ideal binary diffusion system.[2] This derivation was the approach taken by Darken in his original 1948, though shorter methods can be used to attain the same result.

Darken's second equation relates the chemical diffusion coefficient, ${\displaystyle {\tilde {D}}}$, of a binary system to the atomic fractions of the two components. Similar to the first equation, this equation is applicable when the system does not undergo a volume change. This equation also only applies to multicomponent systems, including binary systems, that obey the equations of state and the Gibbs–Duhem equations.

To derive Darken's second equation the gradient in Gibb's chemical potential is analyzed. The gradient in potential energy, denoted by F₂, is the force which causes atoms to diffuse.[2] 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.[6] In addition, N_A is Avogadro's number, and C₂ is the concentration of diffusing component two. This yields

${\displaystyle J=-{\frac {1}{N_{A}}}{\frac {dF_{2}}{dx}}B_{2}C_{2},}$

which can be equated to the expression for Fick's first law:

${\displaystyle -D_{2}{\frac {dC_{2}}{dx}},}$

so that the expression can be written as

${\displaystyle D_{2}{\frac {dC_{2}}{dx}}={\frac {1}{N_{\text{A}}}}{\frac {dF_{2}}{dx}}B_{2}C_{2}.}$

After some rearrangement of variables the expression can be written for D₂, the diffusivity of component two:

${\displaystyle D_{2}={\frac {dF_{2}}{dC_{2}}}{\frac {B_{2}C_{2}}{N_{\text{A}}}}.}$

Assuming that atomic volume is constant, so C = C₁ + C₂,

${\displaystyle {\frac {1}{N_{\text{A}}}}{\frac {dF_{2}}{dN_{2}}}B_{2}N_{2}.}$

Using a definition activity, ${\displaystyle dF_{2}=RT\,d\ln a_{2}}$, where R is the gas constant, and T is the temperature, to rewrite the equation in terms of activity gives

${\displaystyle D_{2}=kTB_{2}{\frac {d\ln a_{2}}{d\ln N_{2}}}.}$

The above equation can be rewritten in terms of the activity coefficient γ, which is defined in terms of activity by the equation ${\displaystyle \gamma _{2}=a_{2}/N_{2}}$. This yields

${\displaystyle D_{2}=kTB_{2}\left(1+N_{2}{\frac {d\ln \gamma _{2}}{d\ln N_{2}}}\right).}$

The same equation can also be written for the diffusivity of component one, ${\displaystyle D_{1}=kTB_{1}\left(1+N_{1}{\tfrac {d\ln \gamma _{1}}{d\ln N_{1}}}\right)}$, and combining the equations for D₁ and D₂ gives the final equation:[2]

${\displaystyle {\tilde {D}}=(N_{1}D_{2}+N_{2}D_{1}){\frac {\partial \ln a_{1}}{\partial \ln N_{1}}}.}$