...
The parameter Q1trans, which is seen to have dimensions of energy, is termed the heat of transport
Mass diffusion can be induced by gradients in either the composition, or the temperature, or both. The origin of Q1trans is the asymmettry between the energy states before, during, and after a diffusing species jumps to a neighboring sitebe determined by finding expressions for the atom flux and the diffusion equation in the crystal, and then solving the diffusion equation subject to the boundary conditions at the surface.
Methods of measuring Q1trans are similar to those for measuring Beta in an electromigratino experimentelectromigration experiment
3.4 Mass Diffusion Motivated by Capillarity
The diffusion potentials of the components in the direct vicinity of an interface depend on the local interface curvature
A simple example is a pure crystalline material with an undulating surface in which self-diffusion takes place by the vacancy exchange mechanism.
There tends to be a diffusion current through the bulk from the convex region to the concave region, smoothing the surface and reducing the total interfacial energy.
The rate of surface smoothing can
3.4.1 The Flux Equation and Diffusion Equation
The system contains two network-constrained components--host atoms and vacancies; the crystal is used as the frame for measuring the diffusional flux, and the vacancies are taken as the Ncth component.
An expression of the coefficients LAA may be obtained by considering diffusion in a very large crystal with flat surfaces.
Find the diffusion equation for vacancies in the absence of significant dislocation sources or sinks within the crystal.
In general, the surface acts as an efficient source or sink for vacancies and the equilibrium vacancy concentration will be maintained in its vicinity.
The vancy concentration far from the surface will generally be a function of the total surface curvature. In this case, the crystal can be assumed to be a large block possessing surfaces which on average have zero curvature. The vacancies in the deep interior can then be assumed to be in equilibrium with a flat surface.
During surface smoothing, differences in the local equilibrium values of Xv maintained in the different regions and differences in vacancy concentration throughout the crystal will be relatively small.
When a vacancy is added to the crystal at a convex region, the crystal expands and the surface area increases. Work must therefore be done to create the additional area.
Because only relatively small variations in cv in typical specimens undergoing sintering and diffusional creep, we prefer to carry out the analyses of surface smoothing, sintering, and diffusional creep in terms of atom diffusion and the diffusion potential. In this approach, the boundary conditions on PhiA can be expressed quite simply. induced by gradients in either the composition, or the temperature, or both. The origin of Q1trans is the asymmettry between the energy states before, during, and after a diffusing species jumps to a neighboring site.
3.4.2 Boundary Conditions
The diffusion potential for the atoms is the surface work term plus the usual chemical term.
The diffusion potential at the convex region of the surface is greater than that at the concave region, and atoms therefore diffuse to smooth the surface
3.5 Mass Diffusion in the Presence of Stress
Because stress affects the mobility, the diffusion potential, and the boundary conditions for diffusion, it both induces and influences diffusion.
3.5.1 Effect on Stress on Mobilities
Consider the diffusion of small interstitial atoms among the interstices between large host atoms in an isothermal unstressed crystal.
The diffusion is isotropic and the mobility is a scalar.
If a general uniform stress field is imposed on the material, no force will be exrted on a diffusing interstitial because its energy is independent of position. The flux remains linearly related to the gradient of the chemical potential. However, M1 will be a tensor because the stress will cause differences in the rates of atomic migration in different directions.
There will be a distortion of the host lattice when the jumping atom squeezes its way from one interstitial site to another, and work must be done during the jump against any elements in the stress field that resist this distortion. Jumps in different directions will cause different distortions in the fixed stress field, so different amounts of work, W, must be done against the stress field during these jumps.
The overall interstitial mobility will be the result of the interstitials making numbers of different types of jumps in different directions.
The mobility should vary linearly with stress and be expressible as a tensor.
3.5.2 Stress as a Driving Force for Diffusion: Formation of Solute-Atom Atmosphere around Dislocations
In a system containing a nonuniform stress field, a diffusing particle generally experiences a force in a direction the reduces its interaction energy with the stress field.
To find the force exerted on an interstitial by a stress field, one must consider the entropy production in a small cell embedded in the material
The diffusion potential is an "elastochemical" type of potential
The flux has two components: the first results from the concentration gradient and the second from the gradient in hydrostatic stress.
In the case where an edge dislocation is suddenly introduced into a region of uniform interstitial concentration, solute atoms will immediately begin diffusing toward the tensile region of the dislocation due to the pressure gradient alone. However, opposing concentration gradients build up, and eventually a steady-state equilibrium solute atmosphere, known as a Cotrell atmosphere, is created where the composition-gradient terms cancels the stress-gradient term.
The first quantifiable theory for the strain aging caused by solute pinning of dislocations.
3.5.3 Influence of Stress on the Boundary Conditions for Diffusion: Diffusional Creep
In a process termed diffusional creep, the applied stress establishes different diffusion potentials at various sources and sinks for atoms in the material. Diffusion currents between these sources and sinks are then generated which transport atoms between them in a manner that changes the specimen shape in response to the applied stress.
Consider a wire specimen possessing a "bamboo" grain structure subject to an applied tensile force. This force subjects the transverse grain boundaries to a normal tensile stress and therefore reduces the diffusion potential at these boundaries.
When the applied force is sufficiently large that the diffusion potential at the transverse boundaries becomes lower than that at the surface, atoms will diffuse from the surface to the transverse boundaries thereby causing the specimen to lengthen in response to the applied stress.
Vacancy fluxes develop in response to gradients in diffusion potential and cause the edge dislocations to climb, and as a result, the wire lengthens in the applied tensile stress direction.
The problem of determining the elongation rate in both cases is therefore reduced to a boundary-value diffusion problem where the boundary conditions at the sources and sinks are determined by the inclination of the sources and sinks relative to the applied stress and the magnitude of the applied stress.
During diffusional creep, the stresses are relatively small, so variations in the vacancy concentration throughout the specimen will generally be small and can be ignored. Quasi-steady-state diffusion may be assumed.
The boundaries will be under traction and when an atom is inserted, the tractions will be displaced as the grain expands.
The displacement coomstributes to work and reduces the potential energy of the system by a corresponding amount.
An increase in the applied force increases sigmann and when sigmann is sufficiently large, atoms will diffuse from the surface of the boundaries at a quasi-steady rate.
3.5.4 Summary of Diffusion Potentials
The diffusion potential is the generalized thermodynamic driving force that produces fluxes of atomic or molecular species. The diffusion potential reflects the change in energy that results from the motion of a species. Following are examples of what the diffusion potentials may include.
- chemical interactions and entropic effects
- network constraint when sites are conserved
- charge in an electrostatic potential
- work against a hydrostatic pressure to move a species with volume
- work against capillary pressure to move a species with volume
- anisotropic equivalent to capillary pressure
- work against an applied normal traction
- change in energy as a dislocation with Burgers vector and unit tangent climbs with stress
- gradient-energy term in the diffuse interface model for conserved order parameters