Chapter 3: Driving Forces and Fluxes for Diffusion
Fluxes of chemical components arise from several different types of driving forces.
The mechanisms of diffusion comprising the microscopic basis for D are essentially independent of the driving force.
All the driving forces can be collected and attributed to a generalized diffusion potential.
Components do not always diffuse independently. There is an introduction of different types of diffusion coefficients defined in specified reference frames to distinguish different diffusion systems.
3.1 Diffusion in Presence of a Concentration Gradient
The diffusion potential of a component is the chemical potential if a concentration gradient exists in a single phase at uniform temperature that is free of all other fields and interfaces.
Gradient in a potential is the driving force of diffusion, and the diffusion flux is proportional to the diffusion potential gradient.
Express the chemical-potential gradient in terms of a concentration gradient.
The factor coupling the flux and concentration gradient is termed diffusivity
The flux is specified relative to a particular reference frame
3.1.1 Self-Diffusion: Diffusion in the Absence of Chemical Effects
A component diffuses in a chemically homogeneous medium during self-diffusion.
Measure with tracer isotopes or marker atoms.
Consider a crystal where self-diffusion takes place by the vacancy-exhange mechanism
Equations are derived from the vacancy-exchange mechanism: every forward jump to an atom occurs via a backward jump of a vacancy.
The self-diffusion of radioactive tracers obeys Fick's law of self-diffusivity designated by *D
3.1.2 Self-Diffusion of Component i in a Chemically Homogeneous Binary Solution
Consider self-diffusion of an isotopic species in a chemically homogeneous binary solution consisting of atoms of types 1 and 2 in the presence of a concentration gradient of the isotope.
Because the crystal remains fixed during the diffusion, the C-frame is used to measure flux.
A Fick's-law expression is obtained for the self-diffusion of the radioactive component.
Fick's first law is used in steady-state diffusion, i.e., when the concentration within the diffusion volume does not change with respect to time (Jin = Jout).
J = - D d phi / d x
D is proportional to the velocity of the diffusing particles, which depends on the temperature, viscosity of the fluid and the size of the particles according to the Stokes-Einstein relation.
3.1.3 Diffusion of Substitutional Particles in a Chemical Concentration Gradient
Consider a solute of type i diffusing on substitutional sites in an inhomogeneous binary solution.
Both the solute particles and host particles interdiffuse on the substitutional sites.
If one species diffuses more quickly than the other, the region initially richer in that species loses net mass and contracts.
The region inititally richer in the more slowly diffusing species gains net mass and expands.
This process is known as the Kirkendall effect after E. Kirkendall.
The Kirkendall effect is the migration of markers that occurs when markers are placed at the interface between an alloy and a metal, and the whole is heated to a temperature where diffusion is possible; the markers will move towards the alloy region. For example, using molybdenum as a marker between copper and brass (a copper-zinc alloy), molybdenum atoms will migrate towards the brass. This is explained by assuming that the zinc diffuses more rapidly than the copper, and thus diffuses out of the alloy down its concentration gradient. Such a process is impossible if the diffusion is by direct exchange of atoms.
From Kirkendall Effect
Description of the Diffusion in a Local C-Frame
Consider interdiffusion. Local planes move with a velocity, v
, with respect to the ends of the sample.
Consider diffusion fluxes in the diffusion zone with respect to its local C-frame. The constraint condition associated with vacancy mechanism requires that fluxes of particles and vacancies sum to zero.
In the Kirkendall effect, the difference in the fluxes of two substitutional species requires a net flux of vacancies.
Vacancy creation and destruction can occur by means of dislocation climb.
Dislocations can slip in planes containing both the dislocation and the Burger's Vector. For a screw dislocation, the dislocation and the Burger's vector are parallel, so the dislocation may slip in any plane containing the dislocation. For an edge dislocation, the dislocation and the Burger's vector are perpendicular, so there is only one plane in which the dislocation can slip. There is an alternative mechanism of dislocation motion, fundamentally different from slip, that allows an edge dislocation to move out of its slip plane, known as dislocation climb. Dislocation climb allows an edge dislocation to move perpendicular to its slip plane.
The driving force for dislocation climb is the movement of vacancies through a crystal lattice. If a vacancy moves next to the boundary of the extra half plane of atoms that forms an edge dislocation, the atom in the half plane closest to the vacancy can "jump" and fill the vacancy. This atom shift "moves" the vacancy in line with the half plane of atoms, causing a shift, or positive climb, of the dislocation. The process of a vacancy being absorbed at the boundary of a half plane of atoms, rather than created, is known as negative climb. Since dislocation climb results from individual atoms "jumping" into vacancies, climb occurs in single atom diameter increments.
During positive climb, the crystal shrinks in the direction perpendicular to the extra half plane of atoms because atoms are being removed from the half plane. Since negative climb involves an addition of atoms to the half plane, the crystal grows in the direction perpendicular to the half plane. Therefore, compressive stress in the direction perpendicular to the half plane promotes positive climb, while tensile stress promotes negative climb. This is one main difference between slip and climb, since slip is caused by only shear stress.
One additional difference between dislocation slip and climb is the temperature dependence. Climb occurs much more rapidly at high temperatures than low temperatures due to an increase in vacancy motion. Slip, on the other hand, has only a small dependence on temperature.
From dislocation climb
Vacancy destruction occurs when atoms from the extra planes associated with dislocations fill the incoming vacancies and the extra planes shrink.
Extra planes expand as atoms are added to them in order to form vacancies.
Contraction and expansion causes mass flow revealed by the motion of embedded inert markers.
Net flux of substitutional atoms across the interface plane result in local volume change.
Dimensional changes parallel to the interface are restricted, and in-plane compatibility stresses are generated.