Materials Science on CDROM 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
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=DHTDS is known.
The module comprises 3 sections:
There is also an appendix to this document.
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.
The student is first taken through a derivation of Fick's 1st law, as it applies
to interstitial diffusion:

(1) 
where J_{B} is the atomic flux of interstitial B atoms (atoms m^{2}
s^{1}), D_{B} is the diffusivity (m^{2} s^{1})
and ¶C_{B}/¶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 steadystate
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.
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 temperaturedependent. It is shown stepbystep how D is related to
temperature via the expression:

(2) 
where D_{0} is the frequency factor and Q_{ID} is
equivalent to the enthalpy of interstitial atom migration, DH_{m}.
Both these terms can be taken as material constants.
A graphical multiplechoice 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 aFe. 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’).
A derivation for Fick's 2nd law is given. This applies to non steadysate conditions,
i.e. those in which interstitial concentration, C_{B} varies with time. The
general form of Fick's 2nd law is given by:

(3) 
For cases in which D_{B }is independent of composition, or where the
ranges of composition are very small, this reduces to:

(4) 
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 
Carburisation 
C_{S} =
Surface concentration
C_{0} = Initial bulk concentration 
Decarburisation 
C_{0} =
Initial bulk concentration 
Diffusion Couple 
C_{1}
= Concentration of steel 1
C_{2} = Concentration of steel 2 
Homogenisation 
C_{mean}
= Mean concentration
b_{0} = Initial concentration amplitude
l = halfwavelength 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
sidebranches from the relevant pages. See appendix at the end of this chapter for a table
and plot of the error function.
This section looks at selfdiffusion of atoms in a singlecomponent 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, X_{v}. (the equilibrium vacancy concentration
is given by ). By substituting this
expression into that for G, the student is shown stepbystep that for selfdiffusion, the
diffusivity is given by:

(5) 
This is very similar to the equation for interstitial diffusion, except that here, Q_{SD},
the activation enthalpy for selfdiffusion includes both vacancy migration and
formation enthalpy terms. An exercise is included to illustrate the magnitudes of
distances covered during selfdiffusion. These are naturally very much smaller than for
interstitial diffusion.
The focus then switches to the experimental determination of selfdiffusion 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 D_{0} and Q_{SD}, the can
then be estimated. Stepby step instructions for the whole experiment are provided
onscreen. Note that the experiments are best carried out in conjunction with a
spreadsheet application such as Excel, Quattro Pro, Lotus 123, etc.
It is observed that for materials of a given crystal structure and bond type, the
values of Q_{SD} 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, D_{v}
and atoms D_{A} are related by the expression:

(6) 
where X_{v} is the vacancy concentration.
In this section, a hypothetical binary alloy system AB 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.
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
Brich side at a different rate to that of B atoms into the Arich side of the couple.
This imbalance in atomic movement must be offset by an equal movement, or drift of
vacancies in the opposite direction.
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
halfplane. 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.
The following expression for the lattice drift velocity, v is derived:

(7) 
where D_{A} and D_{B} are the diffusion coefficients of A
and B respectively and X_{A} 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
or,

(8) 
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:

(9) 
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.:

(10) 
Similarly, the general form of Fick's second law becomes:

(11) 
or for cases where is independent of concentration,

(12) 
Diffusion Coefficients in Substitutional Alloys
For substitutional diffusion in a binary alloy system AB, the diffusivities of each
species, D_{A} and D_{B} are not only a function of
temperature, but also of composition. This is largely due to the variation in vacancy
concentration, X_{v} 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 X_{v},
it follows that D_{A}, D_{B} and the interdiffusion
coefficient, will be compositiondependent.
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 stepbystep that the interdiffusion coefficient can be
obtained by graphical construction for different compositions, C using the
equation:

(13) 
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.
Simulations
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 2dimensional
hexagonal array of 60x30 atoms is used to represent a closepacked 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. G_{A} as a % of G_{B}) 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 G_{A} = G_{B}).
 This imbalance is compensated by an equal and opposite drift of vacancies.
 The drift of vacancies leads to nonequilibrium 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 userdefined 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 (AA, AB
and BB) 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 AA and BB are less than AB, 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 (AA ~ BB < AB)
 'O'  ordering (AA ~ BB > AB)
 'R'  randomising (AA ~ BB ~ AB)
Bibliography
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, McGrawHill, 1953
ReedHill, R.E., Physical Metallurgy Principles, 2nd ed., Van Nostrand,
1973
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 
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 
