Darken's Equations
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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: : \nu = (D_1 - D_2) \frac = (D_2 - D_1) \frac. where: * \nu is the ''marker velocity'' of inert markers showing the diffusive flux. * D_1 and D_2 are the diffusion coefficients of the two components. * N_1 and N_2 are the atomic fractions of the two components. * x 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: :\tilde = (N_1 D_2 + N_2 D_1) (1+N_1\frac). where: * a_1 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. * \tilde 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 \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 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 ...
, -D_1 \tfrac, 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 ...
, 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.


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: :-D_1 \frac. 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 :-D_1 \frac. 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, \nu is the velocity of the markers. :-\left _1\frac - 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 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: : \frac = \frac \left _1 \frac - C_1 \nu\right The same equation can be written for the other component, designated as component two: : \frac = \frac \left _2 \frac - C_2 \nu\right 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: :C = C_1 + C_2. The above equation can then be used to combine the expressions for \tfrac and \tfrac to yield :\frac = \frac \left _1 \frac + D_2 \frac - C \nu\right Since ''C'' is constant, the above equation can be written as :0 = \frac \left _1 \frac + D_2 \frac - C \nu\right The above equation states that \textstyle D_1 \frac + D_2\frac - C \nu is constant because the derivative of a constant is equal to zero. Therefore, by integrating the above equation it is transforms to \textstyle D_1 \frac + D_2 \frac - C \nu = I , where 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. These conditions then allow the equation to be rearranged to give :\nu = \frac \left _1 \frac + D_2 \frac\right Since ''C'' is assumed to be constant, \textstyle \frac = -\frac. Rewriting this equation in terms of atom fraction N_1 = \tfrac and N_2 = \tfrac yields :\nu = (D_1 - D_2) \frac = (D_2 - D_1) \frac.


Accompanying derivation

Referring back to the derivation for Darken's first equation, \nu is written as :\nu = \frac \left _1 \frac + D_2 \frac\right Inserting this value for \nu in \textstyle \frac = \frac \left _1 \frac - C_1 \nu\right/math> gives : \frac = \frac \left _1_\frac_-_\frac_\left[D_1_\frac_+_D_2\frac\rightright.html" ;"title="_1_\frac_+_D_2\frac\right.html" ;"title="_1 \frac - \frac \left[D_1 \frac + D_2\frac\right">_1 \frac - \frac \left[D_1 \frac + D_2\frac\rightright">_1_\frac_+_D_2\frac\right.html" ;"title="_1 \frac - \frac \left[D_1 \frac + D_2\frac\right">_1 \frac - \frac \left[D_1 \frac + D_2\frac\rightright As stated before, \textstyle \frac = -\frac, which gives : \frac = \frac \left[\frac D_1 \frac - \frac \left[D_1 \frac - D_2 \frac\right]\right]. Rewriting this equation in terms of atom fraction N_1 = \tfrac and N_2 = \tfrac yields :\frac = \frac \left[(N_2 D_1 + N_1 D_2) \frac\right]. By using \lambda \equiv \tfrac and solving to the form N_1 = f(\lambda), it is found that :-\frac \lambda \,dN_1 = d N_2 D_1 + N_1 D_2) \frac Integrating the above gives the final equation: :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 In thermodynamics, the Gibbs–Duhem equation describes the relationship between changes in chemical potential for components in a thermodynamic system: :\sum_^I N_i \mathrm\mu_i = - S \mathrmT + V \mathrmp where N_i is the number of moles of comp ...
. This equation, as well as Darken's first law, \nu = (D_2 - D_1) \tfrac, gives a complete description of an ideal binary diffusion system. 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


Background

Darken's second equation relates the chemical diffusion coefficient, \tilde, 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 equation In thermodynamics, the Gibbs–Duhem equation describes the relationship between changes in chemical potential for components in a thermodynamic system: :\sum_^I N_i \mathrm\mu_i = - S \mathrmT + V \mathrmp where N_i is the number of moles of comp ...
s.


Derivation

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. To begin, the
flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
''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.Gaskell, David R. ''An Introduction to: Transport Phenomena in Materials Engineering''. 2nd ed. New York; Momentum Press, 2012. In addition, ''N''A is the
Avogadro constant The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining con ...
, and ''C''2 is the concentration of diffusing component two. This yields : J = -\frac \frac B_2 C_2, which can be equated to the expression for Fick's first law: : -D_2 \frac, so that the expression can be written as : D_2 \frac = \frac \frac B_2 C_2. After some rearrangement of variables the expression can be written for ''D''2, the diffusivity of component two: :D_2 = \frac \frac. Assuming that atomic volume is constant, so ''C'' = ''C''1 + ''C''2, :\frac \frac B_2 N_2. Using a definition activity, dF_2 = RT\,d\ln a_2, where ''R'' is the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
, and ''T'' is the temperature, to rewrite the equation in terms of activity gives :D_2 = kT B_2 \frac. The above equation can be rewritten in terms of the activity coefficient γ, which is defined in terms of activity by the equation \gamma_2 = a_2/N_2. This yields :D_2 = kT B_2 \left(1 + N_2 \frac\right). The same equation can also be written for the diffusivity of component one, D_1 = kT B_1 \left(1 + N_1 \tfrac\right), and combining the equations for ''D''1 and ''D''2 gives the final equation: :\tilde = (N_1 D_2 + N_2 D_1) \frac.


Applications

Darken’s equations can be applied to almost any scenario involving the diffusion of two different components that have different diffusion coefficients. This holds true except in situations where there is an accompanying volume change in the material because this violates one of Darken’s critical assumptions that atomic volume is constant. More complicated equations than presented must be used in cases where there is
convection Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convec ...
. One application in which Darken’s equations play an instrumental role is in analyzing the process of diffusion bonding.Orhan
N , M Aksoy, and M Eroglu. "A new model for diffusion bonding and its application to duplex alloys." Materials Science and Engineering 271.1-2 (1999): 458-468. Science Direct. Web.
Diffusion bonding Diffusion bonding or diffusion welding is a solid-state welding technique used in metalworking, capable of joining similar and dissimilar metals. It operates on the principle of solid-state diffusion, wherein the atoms of two solid, metallic surfac ...
is used widely in manufacturing to connect two materials without using adhesives or welding techniques. Diffusion bonding works because atoms from both materials diffuse into the other material, resulting in a bond that is formed between the two materials. The diffusion of atoms between the two materials is achieved by placing the materials in contact with each other at high pressure and temperature, while not exceeding the melting temperature of either material. Darken’s equations, particularly Darken’s second equation, come into play when determining the diffusion coefficients for the two materials in the diffusion couple. Knowing the diffusion coefficients is necessary for predicting the flux of atoms between the two materials, which can then be used in numerical models of the diffusion bonding process, as, for example, was looked at in the paper by Orhan, Aksoy, and Eroglu when creating a model to determine the amount of time required to create a diffusion bond. 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. Application of Darken’s first equation has important implications for analyzing the structural integrity of materials. Darken’s first equation, \textstyle v=(D_2-D_1)\frac, can be rewritten in terms of vacancy flux, \textstyle J_v=(D_2-D_1)\frac. 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. This is particularly important in materials such as aluminum nickel superalloys that are used in jet engines, where the structural integrity of the materials is extremely important. Porosity formation, known as Kirkendall porosity, in these nickel-aluminum superalloys have been observed when diffusion bonding has been used.Janssen
M.M.P.. "Diffusion in the nickel-rich part of the Ni−Al system at 1000° to 1300°C; Ni3Al layer growth, diffusion coefficients, and interface concentrations." Metallurgical Transactions 4.6 (1973): 1623-1633.Springer Link. Web.
It is important then to use Darken’s findings to predict this porosity formation.


See also

* Gibbs-Duhem equation#Ternary and multicomponent solutions and mixtures


References

{{reflist Binary systems Chemical kinetics Physical chemistry