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Materials Science on CD-ROM User Guide

Atomic Diffusion in Metals and Alloys

Version 2.1

Andrew Green, MATTER
John Humphreys, UMIST/University of Manchester
Ross Mackenzie, Open University
Ian G Jones, MATTER

Assumed Pre-knowledge

Before starting this module, it is assumed that the student is familiar with the crystalline nature of metals and alloys, and understands the terms interstitial, vacancy, substitutional solid solution, binary alloy. References are made to the equilibrium vacancy concentration and it is preferable that this concept had been encountered, although backup is available both through the glossary and via hyperlinks to other modules. Solutions to Fick's laws are stated where applicable, but the mathematical derivations (most of which require a knowledge of partial differential equations) are not given. It is also recommended that the user has at least a basic understanding of metallurgical thermodynamics, and that the Gibbs’ free energy relationship DG=DH-TDS is known.

Module Structure

The module comprises 3 sections:

There is also an appendix to this document.

Interstitial Diffusion

This section concentrates on diffusion in a dilute interstitial solid solution. This is a relatively simple starting point as it assumes that every interstitial atom is surrounded by vacant interstitial sites into which they can jump.

Fick's 1st law

The student is first taken through a derivation of Fick's 1st law, as it applies to interstitial diffusion:


where JB is the atomic flux of interstitial B atoms (atoms m-2 s-1), DB is the diffusivity (m2 s-1) and CB/x is the concentration gradient (atoms m-4).

The key point is made that diffusive flux is proportional to the concentration gradient. An exercise is given in which the user is shown a series of hypothetical concentration profiles (plots of C versus distance, x) and asked how the flux varies with distance.

It is emphasised that Fick's 1st law can only be used to solve steady-state diffusion problems. An example is given of hydrogen gas ‘leaking’ through a steel pipeline, in which it is assumed that the inner and outer surface concentrations are constant.

Diffusivity, D (Also Coefficient of Diffusion)

Before going on to look at Fick's 2nd law, a more detailed insight into the term diffusivity is given. This is explained in terms of the atomic jump frequency, G, which is highly temperature-dependent. It is shown step-by-step how D is related to temperature via the expression:



where D0 is the frequency factor and QID is equivalent to the enthalpy of interstitial atom migration, DHm. Both these terms can be taken as material constants.

A graphical multiple-choice exercise asks the student to identify the correct graphical relationship between D and T before going on to look at plots of D vs. T for some interstitial elements in a-Fe. A further exercise encourages the user to consider the type of distances involved in interstitial diffusion. It also highlights the difference between total and net diffusion distances (i.e. as given by a ‘random walk’).

Fick's 2nd Law

A derivation for Fick's 2nd law is given. This applies to non steady-sate conditions, i.e. those in which interstitial concentration, CB varies with time. The general form of Fick's 2nd law is given by:



For cases in which DB is independent of composition, or where the ranges of composition are very small, this reduces to:



Solutions to Fick's 2nd Law

The section is completed with 4 example solutions to Fick's 2nd law: carburisation, decarburisation, diffusion across a couple and homogenisation. The solutions given are as follows:

Process Solution

CS = Surface concentration

C0 = Initial bulk concentration


C0 = Initial bulk concentration

Diffusion Couple

C1 = Concentration of steel 1

C2 = Concentration of steel 2


Cmean = Mean concentration

b0 = Initial concentration amplitude

l = half-wavelength of cells

t = relaxation time

Note that derivations for these solutions are not provided.

For each of the first 3 solutions, the user is able to change the temperature and see how the concentration profiles (graphs of C versus x) develop with diffusion time.

The homogenisation case is accompanied by animations to illustrate the effect of processing variables (temperature, time) and material condition (initial segregated cell size) have on the concentration profile.

Further information relating to the error function, erf is accessible as side-branches from the relevant pages. See appendix at the end of this chapter for a table and plot of the error function.

Self-Diffusion in Metals

This section looks at self-diffusion of atoms in a single-component crystal. Here, the atoms are only able to move when they are adjacent to one or more vacancies. This is in contrast to interstitial diffusion where the interstitial atoms are nearly always free to move. The jump frequency of an individual metal atom, G is shown to be proportional to the vacancy concentration, Xv. (the equilibrium vacancy concentration is given by ). By substituting this expression into that for G, the student is shown step-by-step that for self-diffusion, the diffusivity is given by:



This is very similar to the equation for interstitial diffusion, except that here, QSD, the activation enthalpy for self-diffusion includes both vacancy migration and formation enthalpy terms. An exercise is included to illustrate the magnitudes of distances covered during self-diffusion. These are naturally very much smaller than for interstitial diffusion.

The focus then switches to the experimental determination of self-diffusion data for materials. The tracer plane method is described, including the relevant solutions to Fick's laws. Having covered the experimental process and analysis, users are then provided with a simulation tracer plane experiment in which they are invited to obtain data for a variety of different metals. By taking measurements of the concentration profiles, it is possible to calculate D for a given temperature. If the process is repeated for different temperatures, values of D0 and QSD, the can then be estimated. Step-by -step instructions for the whole experiment are provided on-screen. Note that the experiments are best carried out in conjunction with a spreadsheet application such as Excel, Quattro Pro, Lotus 1-2-3, etc.

It is observed that for materials of a given crystal structure and bond type, the values of QSD are roughly proportional to the absolute melting temperature. This point is illustrated graphically.

The section is completed by considering how the diffusion of vacancies in a crystal relates to that of atoms. Since vacancies are always free to jump, their jump frequency is much greater than that for atoms. It is shown that the diffusivities of vacancies, Dv and atoms DA are related by the expression:



where Xv is the vacancy concentration.

Diffusion in Substitutional Alloys

In this section, a hypothetical binary alloy system A-B is considered. The student is introduced to the idea that different atomic species generally have a different probability of moving into vacant lattice sites. This gives rise to a number of interesting and important phenomena, which are considered in further detail. These effects are illustrated by means of a diffusion couple between initially pure A and pure B.

Unequal Jump Probability and Vacancy Drift

Firstly, it is shown that if the jump probability of the two species are different, the flux across a given lattice plane will also be different - i.e. A atoms move into the B-rich side at a different rate to that of B atoms into the A-rich side of the couple. This imbalance in atomic movement must be offset by an equal movement, or drift of vacancies in the opposite direction.

Vacancy Creation and Annihilation

It is then shown that as a result of such a drift, the vacancy concentrations on either side of the couple will depart from equilibrium - there will be an excess of vacancies on one side and a depletion on the other. In order to maintain the concentrations at or near equilibrium, vacancies are destroyed or created at various sinks and sources (such as dislocations, grain boundaries, etc.) A series of 3 animations illustrates the phenomenon of vacancy creation and annihilation at edge dislocations. An important result of vacancy creation at an edge dislocation is the extension of the half-plane. Similarly, vacancies sinking into a dislocation cause it to contract. Whole lattice planes can therefore be created or destroyed by vacancy movement, and this gives rise to yet another important effect, that of lattice drift.

Lattice Drift Velocity

The following expression for the lattice drift velocity, v is derived:



where DA and DB are the diffusion coefficients of A and B respectively and XA is the molar fraction of A. The section then goes on to explain that lattice drift will affect the net diffusive flux, , i.e.

Net flux = flux due to concentration gradient + flux due to lattice drift




Darken’s Equations

This leads to the derivation of Darken’s equations (essentially Fick's laws for substitutional diffusion), whereby the net diffusive flux of species A is given by:



Here, is the interdiffusion coefficient, which is a weighted average of the diffusion coefficients of the individual components, with respect to alloy composition, i.e.:



Similarly, the general form of Fick's second law becomes:



or for cases where is independent of concentration,



Diffusion Coefficients in Substitutional Alloys

For substitutional diffusion in a binary alloy system A-B, the diffusivities of each species, DA and DB are not only a function of temperature, but also of composition. This is largely due to the variation in vacancy concentration, Xv with composition. The software illustrates this concept by comparing vacancy concentrations at a given temperature for pure A, pure B and a 50/50 alloy. In general, the metal with the higher melting temperature will contain fewer defects. It is shown that if the two are mixed to form a solid solution, the vacancy concentration for the resulting alloy may take an intermediate value, which will depend on the composition. Since the diffusivity of each species is proportional to Xv, it follows that DA, DB and the interdiffusion coefficient, will be composition-dependent.



Matano Analysis

Since Fick's law cannot be directly integrated for variable, values must be obtained experimentally. An outline is given of the most common method - the Matano analysis. A ‘pure’ diffusion couple is created and annealed at a constant temperature for a given length of time. After removal from the furnace, a concentration profile is generated. From this, the Matano interface is defined as being the plane across which an equal number of atoms have crossed in both directions. It is shown step-by-step that the interdiffusion coefficient can be obtained by graphical construction for different compositions, C using the equation:



The integral term, is the area between the profile and Matano interface, whilst dx/dC is the reciprocal of the curve gradient at C. Some real data is provided in an additional exercise, from which a plot of versus C can be obtained.


The section is completed by 2 graphical simulations of diffusion processes. The first of these shows diffusion across a couple of pure A (red) and B (blue). A 2-dimensional hexagonal array of 60x30 atoms is used to represent a close-packed plane in a crystal. For simplification, the crystal structure and atomic size of both species are identical. Interaction is provided by allowing the user to change the relative jump frequencies of the 2 species(i.e. GA as a % of GB) and observing the effect on the way in which the concentration profile develops. A graph below the atomic array continually updates the ‘interpenetration profile’. Furthermore, the initial number of vacancies on each side of the couple may be set. These numbers are used to approximate ‘equilibrium vacancy concentrations’ on each side of the couple, and these concentrations are automatically maintained at or near ‘equilibrium’ throughout the simulation.

A number of the phenomena described earlier in the section can be observed:

  • Firstly, the interpenetration of A and B atoms across the couple is different, (except when GA = GB).
  • This imbalance is compensated by an equal and opposite drift of vacancies.
  • The drift of vacancies leads to non-equilibrium numbers (or concentrations) on either side of the couple.
  • Vacancies are created on one side of the couple and annihilated on the other. The number of created/destroyed pairs are recorded below the atomic array by the Sink/Source parameter.
  • When this number reaches 30 (the number of atoms/column) then a new atomic plane is drawn on one side and removed from the other. This represents the growth of dislocations acting as vacancy sources and the erosion of those acting as sinks.
  • Lattice drift is thus observed, highlighted by the gradual movement of the single green marker away from its original position at the centre of the array.
  • The lattice drift velocity, v can be estimated by recording the number of jumps every time the green marker moves. The effect of the user-defined parameters on v can be observed by running the simulation a number of times.

Note that these simulations run under separate windows. Instructions on how to pause, resume and terminate the simulations are included in the software.

The second simulation illustrates the effect of relative atomic bond energies (A-A, A-B and B-B) on the microstructural development in a binary system. Again, the crystal structure and atomic size of both species are identical. When run, the simulation displays a 60x30 array of randomly arranged A and B atoms. However, since the default settings are such that the bond energies A-A and B-B are less than A-B, diffusion leads to a clustering of the two components. The following keys can be pressed to alter the nature of the interatomic bonds:

  • 'C' - clustering (A-A ~ B-B < A-B)
  • 'O' - ordering (A-A ~ B-B > A-B)
  • 'R' - randomising (A-A ~ B-B ~ A-B)


The student is referred to the following resources in this module:

Porter, D.A., and Easterling, K.E., Phase Transformations in Metals and Alloys, 2nd ed., Chapman & Hall, 1992

Henderson, B., Defects in crystalline solids, Arnold, 1972

Shewmon, P.G., Diffusion in solids, 2nd ed., TMS, 1989

Flynn, C.P., Point defects and diffusion, Clarendon Press, 1972

Christian, J.W., The theory of transformations in metals and alloys, 2nd ed., Pergamon, 1975

Callister, W.D., Materials science and engineering, 3rd ed., Wiley, 1994

Darken, L.S. and Gurry, R.W., Physical Chemistry of Metals, McGraw-Hill, 1953

Reed-Hill, R.E., Physical Metallurgy Principles, 2nd ed., Van Nostrand, 1973


The Error Function

The error function, erf is given by:

The following table gives values of erf(z) for 0<z<2.8.

z erf(z) z erf(z) z erf(z)
0 0.0000 0.55 0.5633 1.3 0.9340
0.025 0.0282 0.60 0.6038 1.4 0.9523
0.05 0.0564 0.65 0.6420 1.5 0.9661
0.10 0.1125 0.70 0.6778 1.6 0.9763
0.15 0.1680 0.75 0.7111 1.7 0.9838
0.20 0.2227 0.80 0.7421 1.8 0.9891
0.25 0.2763 0.85 0.7707 1.9 0.9928
0.30 0.3286 0.90 0.7969 2.0 0.9953
0.35 0.3794 0.95 0.8209 2.2 0.9981
0.40 0.4284 1.00 0.8427 2.4 0.9993
0.45 0.4755 1.1 0.8802 2.6 0.9998
0.50 0.5205 1.2 0.9103 2.8 0.9999

Matano Interface

Distance (m) Atomic % Cu Distance (m) Atomic % Cu
0 0 0.0010 91
0.0001 1 0.0011 93
0.0002 3 0.0012 95
0.0003 12 0.0013 96
0.0004 33 0.0014 97
0.0005 58 0.0015 98
0.0006 71 0.0016 98
0.0007 80 0.0017 99
0.0008 85 0.0018 99
0.0009 89 0.0019 100


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Copyright of The University of Liverpool 2000