Last Time: Self-Diffusivity
Consider a system with components 1, *1, and v
Ji = Lij Fj
J*1C = -kT [ L11 / C1 - L1*1 / C*1] d*1 / dx
Mix radioactive atoms with non-radioactive atoms. There is a different mass.
Atomic vibrations exert a small effect. The atoms move at slightly different rates
Discuss different types of diffusivities
Analyze data to report diffusion coefficient
Important to distinguish subtleties
Consider the diffusivity with respect to different reference frames
The purpose of this lecture is to discuss main points
Interdiffusion
Interdiffusion occurs when two things are placed together and intermix. Diffusion results from a concentration gradient.
Study diffusion in an ideal system.
Track a given atom and see over a thousand trials where it is.
Find the mean-square displacement and relate to diffusivity
Look at the flux equations of component 1 and *1
The expression looks like Fick's First Law.
There is more detail of what diffusion looks like in thermodynamics.
The term *D is the diffusivity of a radioactive tracer. It is the diffusivity of *1
The self-diffusivity was described. Now extend this to self-diffusivity in an alloy of uniform composition. Keep the terms (N1 + N*1) / (N1 + N*1 + N2) uniform.
Mix in a radioactive component on one side. There is no gradient in the concentration of species 1. Use a similar formula to find almost the same expression.
J*1C = -kT [L11 / C1 - L1*1 / C*1] dC*1 / dx
J*1C = -*D1 dC11 / dx
The self-diffusivity of 1 in the alloy is termed D1*
The subscript indicates that there are other species in the system.
Consider the root-mean-square displacement of *1 moving in a uniform system.
C-Frame of Reference
The c-frame of reference is attached to particular crystal planes.
Measure everything as if the crystal planes remained fixed during motion
Vacancies may be exchanging places, but each plane remains in the same location
Diffusion in a Concentration Gradient
Kirkendall Experiment
Consider the Kirkendall experiment in which there is a diffusion couple of copper and copper-zinc
Tungsten wires are wound around the rod, and everything is electroplated with copper
Look at a section through the rod and the concentration of zinc versus distance
Anneal and allow the two components to interdiffuse
Make concentration versus distance measurements and plot curves
Use x-ray dispersive analysis to measure points and generate a curve
Use SEM to determine where initial markers of tungsten are located
Slopes are shallower on the left than on the right. There is an asymmetry in the diffusion profile. It is shallower on the high zinc side.
Explanation of Observations
There are non-equal fluxes JCu and JZn
Identify a number of self-diffusivity to characterize how fast component 1 would move
The radioactive tracer of 2 would help measure diffusivity. The fluxes of 1 and 2 are not necessarily equal. The flux of copper and zinc may be different.
Assume that JCu + JZn + Jv = 0
Ramifications of Assumption
There is a supply of vacancies on the far right side and vacancy sinks on the left side
A production of vacancies supports the flux difference
There is nothing happening outside the interdiffusion zone.
How Vacancy Sources and Sinks Work
Consider crystal imperfections. Edge dislocation climb is the simplest possible source and sink of vacancies
A dislocation can emit a vacancy
Atoms move to the site at the end of an extra plane and a vacancy migrates away
The climb of the edge dislocation extends the extra half-plane
Vacancies are created by climb
Partial planes become more complete.
Consider entropy of mixing. The equilibrium fraction rises as high as 10^-4. There is a high concentration of vacancies. Equilibrate a piece of copper at 900 C where the vacancy concentration is low. Measure the electrical resistivity with time. The resistivity jumps up, and there is a slow relaxation with additional vacancies. The emission from sources provide vacancies.
Run the source backward to produce sinks
Vacancy sinks cause extra planes to be eroded
Sources extend half planes and planes are locally eliminated with sinks
Application to diffusion couple experiment
Volume contracts on the left side and expands on the right side at the same rate
This results in the movement of markers to the left relative to the end of the sample
The ideal case is that the vacancies are in local equilibrium
Whatever the intrinsix diffusivities are, the system will respond with the correct flux
The production and disruption mechanism occur at the same rate. The total number of sites remain at about the same number.
The atomic planes do not stay in the same position
The middle undergoes a steady migration. There is a local velocity of each plane.
There is a choice of how to describe things
Observe Porosity Forming in the Sample
Porosity may form in the sample
Pores form when the system becomes supersaturated and there is a collection of vacancies. Clusters of vacancies nucleate. Model how they form.
Near the interdiffusion zone is production of porosity. Sinks do not work very efficiently. Porosity forms on one side, and vacancies carry atomic volume
C-Frame Analysis
Flux of 1 in C-Frame
J1C = L11 F1 + L12 F2 + ...
J2C = L21 F1 + L22 F2 + ...
J1C + J2C + JvC = 0
With respect to any plane, the flux is zero. The sum is conserved, and more vacancies move in a certain way.
Assumptions of Local Equilibrium
The gradient of chemical potential is zero in the diffusion zone
There is a flux in vacancies. Must there be a gradient in vacancy concentration?
There is a balance between the production of vacancies and the destruction of vacancies.
The difference between J1 and J2 is accomodated by the flux of vacancies.
The difference between fluxes results in the steady-state flow of vacancies.
There is a response to the difference in concentration of J1 and J2
There are gradients of 1 and 2 but not a gradient of vacancies in interdiffusion zone.
Destroy at a certain rate
There is local equilibrium, and the Gibbs-Duhem equation allows one to simplify.
J1C = -kT [ L11 / C1 - L12 / C2 ][ 1 + d ln gamma / d ln C1 + d ln < omega > / d ln C1 ] dC1 / dx
The activity coefficient is gamma and is related to the thermodynamics of the 1-2 solution
The average atomic volume at a particular concentration is <omega>
There is a thesis of a study of gold-nickel diffusion. Show the value of coefficients
-kT [ L11 / C1 - L12 / C2 ][ 1 + d ln gamma / d ln C1 + d ln <omega> / d ln C1 ]
The intrinsic diffusivity is D1. Everything is concentration dependent.
When <omega> is Independent of Concentration
Consider a gold-nickel phase diagram.
This is not a good approximation to the ideal solution. There is a miscibility gap
The lattice constant of gold is 0.404 and of nickel is 0.357
Expect the average atomic volume not to be independent of volume.
Relate the intrinsic diffusivity to the self-diffusivities
D1 = *D1 [ 1 + d ln gamma1 / d ln C1 ]
The instrinsic diffusivity describes an interdiffusion process. There is a correction due to the thermodynamics of the system. The diffusivities are almost the same.
There is a preference of A-A bonds and B-B bonds at low temperature.
Consider an experiment with an ideal solution. The diffusivity of a radiotracer gold atom is *DAu
Could knowledge of the process of interdiffusion kinetics be applied?
Make up a diffusion couple. What does thermodynamics tend to do?
Does the thermodynamic factor speed up or slow down interdiffusion?
Consider the behavior in an ideal solution and the kinetics in a real material
V-Frame Analysis
Flux is observed in a particular frame.
Consider a volume-fixed frame.
omega1 J1V + omega2 J2V = 0
There is no net flux measured with respect to this frame
Write an expression of the local velocity of the v-frame relative to the c-frame
vCV = (D1 - D2) omega1 dC1 / dx
vCV varies across the diffusion zone.
J1v = -[C1 omega1 D2 + C2 omega2 D1] d C1 / dx
This is a Fick's law type expression.
Note that vCv = 0 at the ends of the sample, and thus the sample ends form the reference frame from which interdiffusion is described (laboratory frame)
When omega1 and omega2 are equal, write the interdiffusivity as below
Dtilda = X2D1 + X1D2
Relate the diffusion coefficients from the c-frame to the single-diffusivity quantities in the v-frame
Darken equation: links the v-frame and c-frame
Summary
V-Frame Description
There is a single term Dtilda related to interdiffusivity
Use a fixed frame of reference, which is often the end of sample
C-Frame Description
There are intrinsic diffusivities of D1 and D2 and a local reference frame attached to crystal planes
The local motion varies through the interdiffusion zone.
Study the motion through the interdiffusion zone.
Consider the diffusion of ink in a river
Ride in a boat with an eyedropper and drop ink into the water. Describe the diffusion of ink outward from the position of the boat. The ink spreads as a Gaussian in the water in all directions.
If the ink is spread from the bank, it is necessary to correct for the fact that the experimental apparatus is carried by the river. The flux causes the appearance of crystal planes moving from one end to another.
The reference frame of the boat is analogous to the c-frame, while the frame from the bank is similar to the v-frame