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.
Consider substitutional binary allow diffusion. The system contains three components, species 1, species 2, and vacancies. Sites can only be created or destroyed at sources
Write the Gibbs-Duhem relation based on the assumption of local equilibrium. A net flux vacancy flux develops in a direction opposite that of the fastest-diffusing species. Nonequilibrium vacancy concentrations would develop in the diffusion zone if they were not eliminated by dislocation climb.
Relate chemical potential gradients to concetration gradients
The local volume expansion arising from the local change of composition contributes to diffusion via the derivative of the average site volume.
Consider the self-diffusivity of species 1 in a chemically homogeneous solution corresponding to *D1. Compare this with the intrinsic diffusivity of the same species in a chemically inhomogeneous solution at the same concentration, corresponding to D1.
The primary difference between D1 and *D1 is a thermodynamic factor involving the concentration dependence of the activity coeffient of component 1.
A thermodynamic factor arises because mass diffusion has a chemical potential gradient as a driving force, but the diffusivity is measured proportional to a concentration gradient and is influenced by the nonideality of the solution.
Describe fluxes by Fick's-law expressions involving two different intrinsic diffusivities, D1 and D2, in a local coordinate system (local C-frame) fixed to the lattice plane through which fllux is measured.
The planes move normal to one another at different rates in a nonuniform fashion due to the Kirkendall effect.
When there is no change in the total specimen volume, the overall diffusion that occurs during the Kirkendall effect can be described in terms of a single diffusivity measured in a single reference frame.
Diffusion in a Volume-Fixed Frame (V-Frame)
To find the volume-fixed V-frame, assume that a frame, designated an R-frame, exists that relates all local C-frames.
There is an equation of the velocity of a local C-frame with respect to the V-frame. It is the velocity of any inert marker with respect to the V-frame.
The interdiffusivity is designated by Dtilda and is related to the intrinsic diffusivities of components 1 and 2
Relate the V-frame to a laboratory frame suitable for experimental purposes. This is provided by the laboratory frame. The ends of the specimen are unaffected by the diffusion and are stationary with respect to each other since there is no change in the overall specimen volume.
The L-frame and V-frame are thus identical.
The measurement of vcv and Dtilda at the same concentration in a diffusion experiment thus produces two relationships involving D1 and D2 and allows their determination. In the V-frame, the diffusion flux of each component is given by a simple Fick's-law expression where the factor that multiplies the concentration gradient is the interdiffusivity D.
In the V-frame, chemical interdiffusion is described by a single diffusivity. In a local C-frame fixed with respect to the local bulk material, the material flows locally with the velocity vcv relative to the V-frame and the description of the fluxes of the two components requires two diffusivities.
The Kirkendall effect alters the structure of the diffusion zone in crystalline materials. The small supersaturation of vacancies on the side losing mass by fast diffusion causes the excess vacancies to precipitate out in the form of small voids, and the region becomes porous.
In 1946, the Kirkendall effect was observed with inert markers in polymer-solvent systems where the large polymer molecules diffused more slowly than the small solvent molecules.
If the osmotic membrane allows rapid diffusion of A but not B, the pressure PAleft and PAright will then relax to equilibrium values until there is no difference in chemical potential across the membrane. This results in a difference in total pressure across the membrane.
If the membrane becomes free to move, it would move to the left, compressing the left chamber and expanding the right to equilibrate the pressure difference. However, if the membrane is constrained, the fluid may cavitate in the left chamber to relieve the low pressure. This is analogous to the formation of voids in the Kirkendall effect.
3.1.4 Diffusion of Interstitial Particles in a Chemical Concentration Gradient
Another system obeying Fick's law is one involving the diffusion of small interstitial solut atoms (componen 1) among the interstitces of a host crystal in the presence of an interstitial-atom concentration gradient. The large solvent atoms (component 2) essentially remain in their substitutional sites and diffuse much more slowly than do the highly mobile solute atoms, which diffuse by the interstitial diffusion mechanism. The solvent atoms may therefore by considered to be immobile.
L11 can be evaluated by introducing the interstitial mobility M1, which is the average drift velocity, v1, gained by diffusing interstitials when a unit driving force is applied.
There is prediction of diffusive flux which depends linearly on the gradient concentration.
The Nernst-Einstein equation expresses a link between the mobility and the diffusivity
3.1.5 On the Algebraic Signs of Diffusivities
The rate of entropy production is nonnegative. M1 is also nonnegative and L11 must be nonnegative.
A negative diffusivity leads to an ill-posed diffusion equation; so formulations based on fluxes and their conjugate driving forces are preferred to Fick's law and are more physical
3.1.6 Summary of Diffusivities
Four different types of diffusivities include the self-diffusivity in a pure material, the self-diffusivity of solute i in a binary system, the intrinsic diffusivity of component i in a chemically inhomogeneous system, and the interdiffusivity in a chemically inhomogeneous system
3.2 Mass Diffusion in an Electrical Potential Gradient
A gradient in electrostatic potential can produce a driving force for the mass diffusion of a species. Two examples of this are the potential-gradient-induced diffusional transport of charged ions in ionic conductors and the electron.
3.2.1 Charged Ions in Ionic Conductors
Consider an ionic material that contains a dilute concentration of positively charged ions that diffuse interstitially.
The conductivity is direcly proportional to the diffusivity
3.2.2 Electromigration in Metals
An applied electrical potential gradient can induce diffusion (electromigration) in metals due to a cross effect between the diffusing species and the flux of conduction electrons that will be present. When an electric field is applied to a dilute solution of interstitial atoms in a metal, there are two fluxes in the system: a flux of conduction electrons, Jq, and a flux of interstitials, J1.
Evaluating the quantity L1q requires understanding the physical mechanism that couples the mass flux of the interstitials to the electron current.
The force arises from the change in the self-consisten electronic charge distribution surrounding the interstitial defect. The defect scatters the current-carrying electrons and creates a dipole, which in turn creates a resistance and a voltage drop across the defect. This dipole, known as Landauer resistivity dipole, exerts an electrostatic force on the nucleus of the interstitial. This current-induced force is usually described phenomenoligically by ascribing an effective charge to the defect, which couples to the applied electric field to create an effective force.
The force, in turn, induces a diffusional drift flux of interstitials.
Consider the interstitial flux in a material subjected to both an electrostatic driving force and a concentration gradient.
Beta can be measured by passing a fixed current through an isothermal system until a quasi-steady state is achieved where J1 approaches zero. Uphill diffusion (flux in the direction of the concentration gradient) takes place until the concentration gradient term cancels the electromigration term.
Electromigration can be used to purify a variety of metals by sweeping interstitials to one end of a specimen
3.3 Mass Diffusion in a Thermal Gradient
Both thermal gradients and electrical-potential gradients can induce mass diffusion.
THe interstitial chemical potential is a function of both concentration and temperature.
The parameter Q1trans, which is seen to have dimensions of energy, is termed the heat of transport
Mass diffusion can be be 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 electromigration 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