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 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.
Methods of measuring Q1trans are similar to those for measuring Beta in an electromigratino experiment