Learning Unit 4:
Diffusion in substitutional solid
solutions
Outline
Introduction
1. Diffusion in an ideal solution, Fick’s first law of diffusion
2. Diffusion in an ideal solution, Fick’s second law of diffusion
3. The Kirkendall effect
4. Porosity
5. The Darken’s equations as essentially Fick’s laws of diffusion in substitutional
solid solutions
6. Solutions to the Fick’s laws of diffusion
7. Solutions to the Fick’s first law of diffusion
8. Solutions to the Fick’s second law of diffusion – Grube method
9. Solutions to the Fick’s second law of diffusion- Matano- Boltzmann method
Outline
10. Determination of the intrinsic diffusivities
11. Self-diffusion in pure metals
12. Temperature dependence of the diffusion coefficient
13. Chemical diffusion at low solute concentration
14. The study of chemical diffusion using radioactive tracers*
15. Diffusion along grain boundaries and free surfaces
16. Fick’s first law in terms of mobility and an effective force*
17. Diffusion in non-isomorphic systems
18. Application of diffusion concept to the homogenization of castings
Objectives
• On completion of this learning unit, you should:
– Understand how diffusion occurs, and what the driving force behind it is
– Understand Fick’s laws and what factors will determine the rate at which
diffusion occurs for some simple cases
– Understand the effects of temperature and microstructure on diffusion
– Be familiar with why diffusion is important to a range of applications
Introduction: diffusion in the context of physical
metallurgy
• Diffusion in solid state as the means to bring in physical contact atoms involved in phase transformations during the
processing/service of metals and alloys
• Diffusion in solid state as the phenomenon that control the kinetics of phase transformations occurring during the
processing/service of metals and alloys
Processing : casting , (cold, warm or
hot) working, metal bonding - welding,
Properties
machining, powder processing, heatheattreatments, surface engineering, etc.
Processing involves phase transformations
Performance
Phase transformation kinetics
• Diffusive processes, Kinetics of diffusion
•Johnson Melh Avrami Kologoromov (JMA/K)
kinetics
•Thermal activation
Structures
Processing
• Competitive nucleation kinetics
• Growth morphologies
• TTT and CCT diagrams
Introduction: Driving force of diffusion –
Down-hill and up-hill diffusion
• Gradient of free energy
• Gradient of chemical potential
• Gradient of activity
• Gradient of concentration
• Gradient of temperature
• Gradient of electric field
• Local state of stress
Diffusion always occurs so as to decrease in Gibbs free energy
(chemical potential).
It could occur down the concentration gradient for down-hill diffusion
(absence of strong interaction between A and B atoms: ideal solid
solution) or up the concentration gradient for up-hill diffusion (gradient
of temperature, presence of strong interaction between A and B atoms
– ordering or clustering systems – spinodal decomposition: negative or
positive deviation, etc.)
Introduction: Diffusion mechanisms (atomic steps)
Substitutional mechanisms : vacancy, direct interchange, cooperative interchange
a.
Vacancy
b.
Interstitial
c.
Interstitial- vacancy
d.
Direct Interchange
e.
Zener- Ring rotation
(cooperative interchange)
! Break of bonds with neighbours and lattice strain →
Energy barrier
The diffusive jump actually requires additional
energy and cooperation from the close-packed
neighbours to accommodate the jumping atom.
Diffusion is thermally activated. It requires an
adjacent vacant site and a high energy of the
solute atom and neighbouring atoms. All these
factors are strongly temperature dependant.
For an active diffusion to occur, the temperature should be high enough to overcome energy barriers to atomic motion.
This energy comes from the thermal energy of atomic vibrations (Eav ~ kBT)
Theoretical computation of energy required of each mechanism, distortion energy and activation energy rather too
complicated compared to experimental determination.
Introduction: Diffusion mechanisms (atomic steps)
Interstitial diffusion is generally faster than vacancy diffusion because bonding of interstitials to the
surrounding atoms is normally weaker and there are many more interstitial sites than vacancy
sites to jump to.
Interstitial diffusion involves small impurity or solute atoms (e.g. C, H, O, N) to fit into the
interstices of the solvent metal.
Introduction:
• Impact of thermal vibration on the formation of vacancies in metals
and alloys
Introduction:
• Simulation of vacancy diffusion mechanism in substitutional solid
solutions
Introduction:
• Self-diffusion in pure metals
Label some atoms
After some time
C
C
A
D
A
D
B
B
No driving force exists, but atoms still diffuse. However, the overall movement of atoms is zero, as the flux of atoms
is the same in every direction.
Introduction:
• Simulation of downhill diffusion
The direction of flow of atoms is opposite the vacancy flow direction.
Introduction:
• Interdiffusion (impurity or solute diffusion) in metals and alloys
Different atomic species generally have a different probability of jumping 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.
Initially
After some time
Adapted from
Figs. 5.1 and
5.2, Callister
7e.
Introduction: Modelling (mathematics) of diffusion
Flux of diffusion
The flux of diffusion, J, is used to quantify how fast diffusion occurs in non-steady
conditions , i.e. when the concentration gradient and the local concentration profile are
changing with time.
The flux of diffusion is defined as the number of atoms crossing a unit cross-section area per
unit time (atoms/m2-second).
J = natom / A t
[atom m-2 s-1]
where natom is the number of atoms crossing the area A during the time t.
Introduction: Modelling (mathematics) of diffusion
Rate of diffusion
The rate of diffusion, ∂n/∂t, is used to quantify how fast diffusion occurs in non-steady
conditions, i.e. when the concentration gradient and the local concentration profile are
changing with time. Its measures the local rate of change of composition with time.
The rate of diffusion is defined as the number of atoms crossing a cross-section area per unit
time (atoms/second).
∂n/∂t = natom/t
[atom s-1]
where natom is the number of atoms diffusing through the area A during time t.
Introduction: Modelling (mathematics) of diffusion
Steady-state diffusion (Fick’s first law)
Flux of diffusion J - the flux of diffusion does not change with time. The J equation does not explicitly
consider the change of concentration with time, the concentration remains constant at each point, or the
significant parameters do not change with time.
Non-steady-state diffusion (Fick’s second law)
Rate of diffusion ∂n/∂t - the flux of diffusion do change with time.
Approach
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.
Validity
The phenomenological description based on the Fick’s laws is valid for any atomic mechanism of
diffusion. Understanding of the atomic mechanisms is important, however, for
predicting the
dependence of the atomic mobility (and, then, diffusion coefficient) on the type of interatomic
bonding, temperature, and microstructure.
Introduction: Modelling (mathematics) of diffusion
Factors that influence diffusion:
Crystal structure
If a crystal structure is distorted either by elastic strains or by extensive plastic deformation, the rate of
diffusion is usually increased.
Another effect of crystal structure is the variation of the diffusion coefficient with crystal direction in a
single crystal of the solvent metal .Such anisotropy is nearly or completely absent in cubic metals, but
bismuth (rhombohedral lattice) shows a ratio of about one thousand in its self-diffusion constants
measured parallel and perpendicular to the c-axis)
Diffusing species, impurities
The presence of small amounts of additional metals usually has a relatively small effect on the diffusion
of solute atoms in a solvent metal (solvent-impurity and solute -impurity interactions).
Frequency factor D0, and activation energy Q are different for every solute, solvent pair.
Diffusion mechanism - Diffusion by interstitial mechanism is usually faster than by substitutional or
vacancy mechanism
Concentration
Temperature - Diffusion rate increases very rapidly with increasing temperature
Microstructure (Grain size and short-circuit paths of diffusion ). Diffusion is faster in polycrystalline
materials compared to single crystals because of the accelerated diffusion along grain boundaries.
1. Diffusion in an ideal solution, Fick’s first law of
diffusion
From experimental, diffusion in FCC, BCC, HCP materials occur by
vacancy mechanism
Random jumps, same probability of jumps – jump rate not function of
concentration.
A as solute, B as solvent,
Solid solution A-B with variation of [A] along the length. [A]right > [A]left
However the [A] is constant throughout the cross-section.
1. Diffusion in an ideal solution, Fick’s first law of
diffusion
Difference of [A] between 2 adjacent transverse atomic planes, in [atom m-4]
Sign “- “ means downhill diffusion, either of A or B atoms
The key point is made that the flux of diffusion is proportional to the
concentration gradient, driving force of diffusion.
Coefficient of diffusion or diffusivity, D, in [m2s-1]
Fick’s 1st law of diffusion
Flux of diffusion of A atoms in [atoms m-2 s-1]
1. Diffusion in an ideal solution, Fick’s first law of
diffusion
Unequal jump probability and vacancy drift
It is shown that if the jump probability of solute A and solvent B 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.
DA ≠ DB → JA ≠ JB
This imbalance in atomic movement must be offset by an equal movement, or drift of vacancies in the opposite
direction.
Vacancy creation and annihilation
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.)
1. Diffusion in an ideal
solution, Fick’s first law of
diffusion
1. Diffusion in an ideal solution, Fick’s first law of
diffusion
1. Diffusion in an ideal solution, Fick’s first law of
diffusion
1. Diffusion in an ideal solution, Fick’s first law of
diffusion
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.
1. Diffusion in an ideal solution, Fick’s first law of
diffusion
Diffusivity D in substitutional solid solutions is a function of T°and composition n, D = f (T°, n)
This is largely due to the variation in vacancy concentration, Nv with composition and T°.
1. Diffusion in an ideal solution, Fick’s first law of
diffusion
In general, the metal with the higher melting T° 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 (image of the interdiffusion
coefficient D, a weighted average of the diffusion
coefficients DA and DB with respect to alloy
composition), 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, D, will be composition-dependent.
2. Diffusion in an ideal solution, Fick’s second law of
diffusion
In many real situations the concentration gradient and the local concentration profile are changing with time.
The changes of the local concentration profile can be described in this case by a differential equation, Fick’s second law.
Fick’s 2nd law of diffusion
2. Diffusion in an ideal solution, Fick’s second law of
diffusion
Solution of the Fick’s 2nd law of diffusion is the local concentration profile nA (x,t):
Fick’s second law relates the rate of diffusion ∂nA/∂t (local rate of change of composition with time) to the curvature
∂2nA/∂x2 of the local concentration profile nA (x,t)
∂2nA/∂x2 > 0 , ∂nA/∂t > 0
∂2nA/∂x2 <0 , ∂nA/∂t <0
The Fick’s 2nd law of diffusion is basically saying : if the local concentration profile is a maximum, it will reduce
with time, wheareas if the profile is a minimum, it will increase with time
Composition nA decays with distance x - An initial step concentration will evolve into a sigmoidal distribution after a
diffusion annealing - The longer the diffusion annealing, or the higher the temperature, the further the profile will penetrate.
3. The Kirkendall effect
•In binary solid solutions, each of the 2 atomic forms can move with a different velocity.
•Gradient of concentration
•Fine wires in refractory material in the plan of weld serve as markers.
•Assumption: metals are pure at the start
•Heating to T°close to melting point of A and B
•Holding for a relative long time at the latter T°, e g for 2 days.
•Subsequent cooling at the ambient T°
•Flow of A atoms from right to left
•Flow of B atoms from left to right
•Interesting result: the wires “move” during the process.
3. The Kirkendall effect
Movement of wires to the right for
distance x.
The only way to explain the movement
of wires is to point out that A atoms
diffuse faster than B atoms.
Kirkendall
effect
=
confirmation
vacancy diffusion mechanism.
Displacement of Kirkendall markers Ξ lattice drift
of
4. Porosity
The element with the low melting point diffuses faster. eg Cu and Ni, Cu diffusion is faster
The right side lost more atoms that it gain, then shrinkage at the right and expansion at the left.
Undersaturation at the right and supersaturation at the left, tensile stress at the right and compressive stress at the left.
Grain boundaries and exterior surfaces are probable positions for both creation and elimination of vacancies.
5. Darken’s equations as essentially Fick’s laws of
diffusion in substitutional solid solutions
The Kirkendall effect shows that in a diffusion couple composed of two metals, the atoms of the components
move at different rates, and the flux of atoms through a cross-section defined by markers is not the same
for both atom forms. It is, thus, more logical to think in terms of diffusivities, DA and DB corresponding to the
movement of the A and B atoms respectively. These quantities may be defined by the following relationships.
DA and DB are function of composition and then of position along diffusion couple. These quantities are valuable
because they measure the speed with which the individual atomic forms move during diffusion
Darken’s equations make it possible to determine the intrinsic diffusivities experimentally.
Assumptions:
•volume contraction and volume expansion only in the length direction
•Cross-section area = constant
•Diffusion zone <<<<<<< diffusion couple
•Porosity does not occur.
5. Darken’s equations as essentially Fick’s laws of
diffusion in substitutional solid solutions
Velocity of Kirkendall markers v - Darken’s first equation
5. Darken’s equations as essentially Fick’s laws of
diffusion in substitutional solid solutions
Velocity of Kirkendall markers Ξ
lattice drift velocity
5. Darken’s equations as essentially Fick’s laws of
diffusion in substitutional solid solutions
Interdiffusion coefficient D - Darken’s second equation
v and ∂nA/∂x could be determined experimentally, but a second equation is compulsory so that to solve
the 2 unknowns DA and DB
Volume mmnn = 1. dx
Rate at which the number of A atoms changes inside mmnn =
Difference in the number of A atoms jumping into and out mmnn per second
Or difference in flux of A atoms across the surface at x and x+dx
5. Darken’s equations as essentially Fick’s laws of
diffusion in substitutional solid solutions
Lattice drift velocity v will affect the net flux of diffusion, JA,
Net flux of diffusion leads to the
→ Net flux = flux due to concentration gradient + flux due to lattice drift
derivation of Darken’s equations,
whereby the net flux of diffusion
of A solute atoms is given by:
Similarly, the net rate of diffusion
of A solute atoms is given by:
5. Darken’s equations as essentially Fick’s laws of
diffusion in substitutional solid solutions
Interdiffusion coefficient
Darken’s equations
6. Solutions to the Fick’s laws of diffusion
Fick's two laws of diffusion are differential equations and must be solved to obtain an algebraic equation describing
the variation of concentration nA as a function of distance x and (usually) of time t.
The ease of solution depends on such factors as the geometry of the specimen in question, the initial
conditions of concentration in the specimen, and the "boundary conditions'" [for example, whether diffusion
penetrates the entire (finite) specimen or affects only a portion of an (infinite) specimen].
The solutions can be listed in three categories:
(1) Those that can be obtained easily by elementary manipulations of the diffusion equations. A majority of
the diffusion problems encountered by an engineer can be simplified to permit at least an order-ofmagnitude solution in this manner.
(2) A variety of technically important diffusion problems have been solved by more advanced mathematical
techniques, and the solutions conveniently collected in reference volumes. Solutions of Fick's second law for
cylinders and spheres are examples.
(3) Diffusion problems of many types, such as those involving concentration-dependent diffusion
coefficients, are solved most conveniently using a digital computer (or perhaps an analog computer).
Examples can be found in technical journals, but usually each special problem must be solved individually.
7. Solutions to the Fick’s first law of diffusion
A majority of the diffusion problems encountered by an engineer can be simplified to permit at least an order-ofmagnitude solution in this manner.
Diffusion of hydrogen through metals is an especially simple example of steady-state diffusion
.
Oxidation of metals is another example of steady-state diffusion.
The treatment of oxidation in the following section illustrates a typical procedure.
7. Solutions to the Fick’s first law of diffusion
• It is assumed that diffusion of only the metal atoms need be considered.
Experimental data on the high-temperature oxidation of metals frequently show a parabolic relation between the time of
oxidation and the amount of oxide formed. This behavior can be explained in terms of a diffusion process.
Furthermore, the difference between the metal concentration NA2 at the oxide surface and the concentration NA1
at the metal surface has a constant value ∆NA.
Therefore, at any thickness x of the oxide layer the flux of diffusion of metal atoms (rate of transfer of metal atoms per
unit area) is: J = - D (∆NA/x) = dm/dt
7. Solutions to the Fick’s first law of diffusion
However, dm and the increase in thickness, dx, are proportional: that is, dx =K dm. Therefore :
dx = - D’ (∆NA/x) dt where D’ =K D and is a constant when D is assumed to be constant.
This equation can be solved as follows:
→
x2 = 2 D’ ∆NA t
x2 = K’ t
K' is positive, since the minus sign cancels the minus sign of the concentration difference ∆NA. The thickness of the
oxide layer increases as the square root of the time of oxidation.
8. Solutions to the Fick’s second law of diffusion –
Grube method
2nd order partial differential equation
Grube method assumption: Diffusivity D very slightly variable, in other words D is constant.
However, the Grube method may be applied to diffusion in alloy systems where the diffusivity varies moderately with
composition if the two halves of a couple are made of alloys differing slightly in composition.
A solution, NA = f (x, t) of this equation must satisfy both the 2nd order partial differential equation and also the initial and
boundary conditions (for space and time) in question.
Examples of solutions to Fick's 2nd law:
Infinite solid, : Diffusion across a couple,
Thin-film solution, thin or finite source:
Deposit of a thin layer of metal A at the end of a long bar of metal B, or in sandwich in between two long bars
of metal B
Semi-infinite solid, constant concentration - infinite source :
Addition of A atoms from a gas phase (diffusion across a semi-couple), e.g. carburisation or case hardening
Removal of A atoms via a gas phase (diffusion across a semi-couple), e.g. decarburisation
Homogenisation (solute equilibration)
Elimination of micro segregation in castings, dissolution of intermetallic compounds
8. Solutions to the Fick’s second law of diffusion –
Grube method
Note that derivations for these solutions are not provided. The solutions given are as follows:
8. Solutions to the Fick’s second law of diffusion –
Grube method
8. Solutions to the Fick’s second law of diffusion –
Grube method
8. Solutions to the Fick’s second law of diffusion –
Grube method
Error function or probability integral function erf (y)
erf (y) becomes negative when y
[or x in x/2(Dt)1/2 ] is negative
8. Solutions to the Fick’s second law of diffusion –
Grube method
More accurate values of erf (y) can be obtained from books of standard mathematical functions
8. Solutions to the Fick’s second law of diffusion –
Grube method
Diffusion distance xd
Since erf (0.5) ≅ 0.5 the depth at which NA is midway between NA1 and NA2 is given by [x/2 (Dt)1/2] ≅ 0.5 ,
that is x ≅ (Dt)1/2
Diffusion distance = xd
≅ (Dt)1/2
Clearly there is no such thing as
a “fixed “diffusion distance”
The diffusion distance xd is just
an estimation of the scale of
diffusion, i.e. an estimation of
the diffusion effects.
The
diffusion
distance
xd
is
yd = xd /2 (Dt)1/2≅ 0.5
approximately the distance over
which the solute will have fallen
to half its initial value.
yd
8. Solutions to the Fick’s second law of diffusion –
Grube method
8. Solutions to the Fick’s second law of diffusion –
Grube method
Theoretical penetration curve
Distance versus composition curve obtained when the solution of Fick's 2nd equation NA is plotted as a function of
the variable y = x/2 (Dt)1/2
Thus, if a diffusion couple has been maintained at some fixed temperature for a given period of time (t) so that
diffusion can occur, then a single determination of the composition at an arbitrary distance (x) from the
weld permits the determination of the diffusivity D
8. Solutions to the Fick’s second law of diffusion –
Grube method
Thus, suppose that a diffusion couple is formed by the welding together of two alloys of the elements A and B
having the compositions NA1, 40 % A (the alloy to the right of the weld), and NA2, 50 % A (left of the weld).
Let the couple be heated quickly to some temperature T1 and held there 40 hr (144 000 sec), and let it be assumed
that after cooling to room temperature, chemical analysis shows that at a distance 2 x 10-3 m to the right of the
weld the composition NA is 42.5 % A
We have on substituting the assumed data in Eq. 12.20
8. Solutions to the Fick’s second law of diffusion –
Grube method
Relationship between distance and time corresponding to the same composition NA during
isothermal diffusion
Since the composition NA remains the same, this problem requires that the argument of the probability integral
has the same value as in the last example. With the assumption of a constant diffusivity (D), that is:
In the numerical computation just considered, a
composition NA = 42.5 % A was assumed at a
distance of 2 x 10-3 m from the weld when the
diffusion time was 40 hr. Compute the length of
time needed to obtain the same composition,
42.5 % A, at twice the distance from the weld
interface.
9. Solutions to the Fick’s second law of diffusion –
Matano- Boltzmann method
Assuming that :
Interdiffusion coefficient D is function of composition D = f (NA),
Diffusion process does not change the composition at the ends of the diffusion couple as boundaries conditions,
Number of atoms per unit volume (nA + nB) is constant and,
No porosity forms at the interface.
Boltzmann proposed the following solution to the Fick’s 2nd law of diffusion
Analytical evaluation of the derivative and integral in Equation 12.24 are much more difficult.
As a consequence, the Matano-Boltzmann method of determining the interdiffusion coefficient D uses graphical
derivation and integration to find out the different factors in Equation 12.24
9. Solutions to the Fick’s second law of diffusion –
The Matano- Boltzmann method
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.
The first step, after the diffusion anneal and chemical analysis of the specimen, is to plot a curve of concentration
versus distance along the bar measured from a suitable point of reference, say from one end of the couple. From
this, the Matano interface is defined as being the plane across which an equal number of atoms have crossed in both
directions.
The second step is to determine that cross-section of the bar through which there have been equal total fluxes of the
two atomic forms (A and B). This cross-section is known as the Matano interface and lies at the position where areas
M and N in Fig. 12.13 are equal.
The position of the Matano interface is determined by graphical integration, but, in general, it has also been
experimentally determined that, in the absence of porosity, the Matano interface lies at the position of the original
weld.
9. Solutions to the Fick’s second law of diffusion –
The Matano- Boltzmann method
Once the Matano interface has been located, it serves as the origin of the x coordinate. Distances to the right of
the interface are considered positive, while those to the left of it are negative.
With the coordinate system thus defined, the Boltzmann solution of Fick's equation is
The integral term, is the area between the penetration curve and Matano interface, whilst ∂x/∂NA is the reciprocal of
the concentration gradient (slope of the penetration curve ) at NA.
With the aid of concentration profile:
The derivative ∂x/∂NA is evaluated by drawing a tangent to the concentration profile at NA and measuring its
slope.
The integral, limits of which are NA1 and NA, is evaluated by a graphical method such as the Simpson's rule.
9. Solutions to the Fick’s second law of diffusion –
The Matano- Boltzmann method
Table 12.3 represents this concentration-distance data corresponding to no actual alloy system, but it is representative,
in a broad general way, of diffusion data obtained in actual experiments. The diffusion couple is assumed to be formed
from pure A and pure B.
Figure 12.14 shows the penetration curve obtained when the data of Table 12.3
are plotted.
9. Solutions to the Fick’s second law of diffusion –
The Matano- Boltzmann method
Let us reconsider the Boltzmann solution of
Fick's 2nd law:
Suppose that we desire to know the
interdiffusion coefficient at a particular
concentration, which we shall arbitrarily
take as NA = 0.375. (point C in Fig. 12.14).
Diffusion time is assumed to be 50 hr
(180,000 s)
∂x/∂NA = reciprocal of the slope of line E tangent to the penetration curve at point C = 1 /(610 m-1) = (1/610) m
Integral (limits of which are NA1 = 0 and NA = 0.375) = cross-hatched area (F) = 4.66 x 10-4 m
9. Solutions to the Fick’s second law of diffusion –
The Matano- Boltzmann method
Computations similar to the above may be made to determine the interdifussion coefficient D(NA) at any
concentration not too close to the terminal compositions (NA = 0 and NA = 1). Since the desired area approaches zero
as the composition approaches one of the terminal compositions, the accuracy of the determination falls off as NA
becomes very close to either 0 or 1.
Large values of D are obtained as one approaches the
concentration NA = 1.
There is a minimum in this curve in the middle of the
concentration range. A diffusivity-concentration curve of this form
has been reported for the diffusion of Zr and U.
However, the curves of Fig. 12.3 are more typical of the diffusion
data reported to date.
10. Experimental determination of the intrinsic
diffusivities
There has been, to date, very little success in the development of a theory capable of predicting the numerical
values of the intrinsic diffusivities starting from a consideration of atomic processes.
While it is generally agreed that diffusion in metallic substitutional solid solutions is the result of the movement of
vacancies, the factors that control jump rates into vacancies of the two atomic forms are complex and not
completely understood.
Thus, in our previous derivation of Fick's first law, several simplifying assumptions were made that do not hold
for real metallic substitutional solid solutions.
First, it was assumed that the solution was ideal, but, as we have seen, most metallic solutions are not ideal, and
in a nonideal solution the diffusion rates are influenced by a tendency for like atoms either to group together, or
to avoid each other (negative and positive deviation).
Second, it was assumed that the rate of jumping was independent of composition, that is, whether the jump was
made by an A or a B atom. The assumption that the rate of jumping is independent of the composition is certainly not
true, as may be judged by the fact that measured diffusivities vary widely with composition.
Experimental determination of intrinsic diffusivities is the viable alternative.
10. Experimental determination of the intrinsic
diffusivities
The determination of the intrinsic diffusivities will now be illustrated with the use of
the assumed data of Table 12.3.
First we must derive an expression for the marker velocity v in terms of the
marker displacement and the time of diffusion t.
Experimentally, it has been determined that the markers move in such a manner
that the ratio of their displacement squared to the time of diffusion is a constant.
Thus
10. Experimental determination of the intrinsic
diffusivities
The flux of A atoms through the marker interface from right to left is
approximately 1.2 times that of the flux of B atoms moving from. left to right.
10. Experimental determination of the intrinsic
diffusivities
11. Self-diffusion in pure metals
In self-diffusion studies, one investigates the diffusion of a solute consisting of a radioactive isotope in a solvent
that is a nonradioactive isotope of the same metal.
In such a system, both atomic forms are identical except for the small mass difference between the isotopes. The
principal effect of this mass difference is to cause the solute isotope to vibrate about its rest point in the lattice with
a frequency slightly different from that of the solvent isotope, giving the two isotopes a slightly different jump
rate.
This difference is easily calculated because the vibration frequency is proportional to the reciprocal of the square
root of the mass, and since the jump rate into vacancies is proportional to the vibration frequency, we have
where 1/T and 1/T* are the jump-rate frequencies of normal and radioactive isotopes respectively (τ and τ* are the mean
times of stay of the respective atoms at lattice positions, and m and m* are the masses of the two isotopes [m*
radioactive]).
11. Self-diffusion in pure metals
Except for the mass difference, solute and solvent are chemically identical and the solid solution is truly ideal.
Considerations of the effect of the departure of a solution from ideality may thus be neglected. Furthermore, the mass
correction is usually small so that, to a good approximation, we may assume that the intrinsic diffusivity of the radioactive
isotope is the same as that of the nonradioactive isotope.
When the intrinsic diffusivities are equal, the interdiffusion coefficient equals the intrinsic diffusivities, as may be seen by
considering the Darken equation:
where D is the interdiffusion coefficient, D = DA = DB, the intrinsic diffusivity of either the radioactive or nonradioactive
isotopes, and by the definition of atom fractions (NA + NB) = 1.
Because the intrinsic coefficients do not depend on composition, it is also true that the interdiffusion coefficient
does not depend on composition.
Therefore experimental determinations of self-diffusivities may be made by using the simpler Grube method.
11. Self-diffusion in pure metals
Because self-diffusion in pure metals occurs as in an ideal solution and with diffusivity that is independent of concentration,
experimentally determined self-diffusion coefficients of pure metals are usually of high accuracy.
Measured diffusivities are capable of theoretical interpretation because the diffusion process occurs in a relatively simple system.
The assumptions made in our derivation of Fick's first law are those actually observed in self-diffusion experiments, and Eq. 12.7 is
correct for self-diffusion in a simple cubic system: i.e.,
11. Self-diffusion in pure metals
Frequency of jumps of an atom in a pure metal crystal
The average thermal energy of an atom (kBT = 0.026 eV for room temperature) is usually much smaller that the
activation energy ∆Hm (~ 1 eV/atom) and a large fluctuation in energy is needed for a jump
ra = Frequency of jumps of an atom = Number of successful jumps made per second by an atom
ra = Jump rate of atoms into vacancies = Number of jumps made per second by an atom
ra = 1/mean time of stay of an atom at a lattice site = 1/τ
ra = (Attempted jumps) x (Probability of a successful jump)
Attempted jumps = attempt frequency = frequency or number of times per second that an atom moves toward
a vacancy = lattice vibration frequency (temperature-independent) = ν
Most of the time the site adjacent to an atom will not be vacant and the jump will not be possible.
Probability of a successful jump = (Probability that an adjacent site is vacant) x (Probability that any attempt
at jumping will be successful)
Probability that an adjacent site is vacant = (Lattice coordination number ) x (Probability that any one site is
vacant)
Lattice coordination number = number of nearest neighbours = Z
Probability that any one site is vacant = molar fraction of vacancies in the metal = Nv = exp (-∆Hf/RT)
∆Hf = enthalpy change or work to form a mole of vacancies
Probability that an adjacent site is vacant = Z exp (-∆Hf/RT)
11. Self-diffusion in pure metals
Probability that any attempt at jumping will be successful = probability that an atom will have sufficient
energy to make a jump = exp (-∆Hm/RT)
∆Hm = enthalpy change or energy barrier that must be overcame to move a mole of atoms into a
mole of vacancies
Probability that any attempt at jumping will be successful = exp (-∆Hm/RT)
Probability of a successful jump = Z exp (-∆Hf/RT) exp (-∆Hm/RT)
ra = ν Z exp (-∆Hf/RT) exp (-∆Hm/RT)
ra = Z ν exp [-(∆Hf+∆Hm)/RT]
ra = 1/τ
1/τ = Z ν exp [-(∆Hf +∆Hm)/RT]
12.31
Equation 12.31 neglects entropy changes associated with the formation and movement of vacancies and should be
more correctly written
11. Self-diffusion in pure metals
Coefficient of self-diffusion D
where ∆Gm and ∆Gf are the free-energy changes associated with the movement and formation of vacancies respectively.
where ∆Sm is the entropy change per mole resulting from the strain of the lattice
during the jumps, and ∆Sf the increase in entropy of the lattice due to the
introduction of a mole of vacancies.
In the BCC lattice, α is 1/8, while Z is 8, and, similarly, in the FCC lattice, α is 1/12, while Z is 12 so that for both forms of
cubic crystals
This expression will be discussed further when we consider the temperature dependence of the diffusion coefficient.
For the present, it suffices to point out that considerable success has been obtained in the theoretical interpretation
of the various factors in this equation.
11. Self-diffusion in pure metals
Self-diffusion in pure metals is essentially viewed as a vacancy diffusion, involving ∆Hf >0. Atoms are only
able to move when they are adjacent to one or more vacancies.
Temperature plays a significant role in the rate of diffusion, as it alters the equilibrium concentration of vacancies, and
probability of a successful jump into a neighbouring site.
Self-diffusion
in pure metals could be viewed as interstitial diffusion by considering ∆Hf = 0. Indeed,
interstitial diffusion is nearly always self-interstitial diffusion. In other words, since the formation of a vacant site
is not needed for interstitial atoms, interstitial atoms are nearly always free to move.
How does 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: Dv = DA /Nv
where Nv is the molar fraction of vacancies in the metal
12. Temperature dependence of the diffusion
coefficient
It has already been seen that the diffusion coefficient is a function of composition. It is also a function of temperature.
The nature of this temperature dependence is shown clearly in the equation for the self-diffusion coefficient stated in
the previous section,
Temperature dependence of the diffusion coefficient D follows the Arrhenius dependence. In this form, the
equation can be applied directly to the study of experimental data.
12. Temperature dependence of the diffusion
coefficient
For interstitial diffusion Q = QID = ∆Hm = enthalpy of interstitial atom migration
For vacancy diffusion Q = QSD =∆Hf + ∆Hm = enthalpy of formation of vacancies + enthalpy of atom migration
Since the formation of a vacant site is not needed for interstitial atoms, Qinterstitial << Qsubstitional
Hence Dinterstitial >> Dsubstitional.
Comparison of diffusion distances in vacancy and interstitial diffusion points out that xinterstitial >> xsubstitional
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.
12. Temperature dependence of the diffusion
coefficient
The slope of the experimentally determined straight line determines the activation energy Q since
m = - Q/2.3 R or Q = -2.3 R m.
At the same time, the intercept of the line with the ordinate designated by b yields the frequency
factor D0, since b = log D0 or D0 = 10b.
12. Temperature dependence of the diffusion
coefficient
The above method of determining experimental activation energies and frequency factors can be illustrated with the use
of some representative data given in Table 12.4.
12. Temperature dependence of the diffusion
coefficient
The experimentally determined
equation for the self-diffusion
coefficient is, accordingly,
12. Temperature dependence of the diffusion
coefficient
The preceding discussion has been concerned only with the temperature dependence of self-diffusion coefficients.
However, experimentally determined values of chemical interdiffusion coefficients D, and of their component
intrinsic diffusivities DA and DB, also show the same form of temperature dependence.
Speaking in general, all diffusion coefficients tend to follow an empirical activation law, so that we have for self-diffusion
13. Chemical diffusion at low solute concentration
14. The study of chemical diffusion using radioactive
tracers*
15. Short-circuit paths of diffusion, diffusion along
grain boundaries and free surfaces
Experimental measurements have shown that the surface and grain-boundary forms of diffusion also obey activation, or
Arhennius-type laws.
Diffusion is, more rapid along grain boundaries than in
the interiors of crystals - free-surface diffusion rates are
larger than either of the other two.
Because the regularity of the crystal lattice of a metal is
disturbed at a grain boundary, near a free surface, and
adjacent to a dislocation line, diffusion by the vacancy
mechanism is greatly enhanced in these regions.
Grain boundaries : open structures containing voids,
dislocations and possessing already a substantial level of
distortions
Both the number of vacancies and their mobility may
be larger as a result of the local disruption of
crystalline regularity.
15. Short-circuit paths of diffusion, diffusion along
grain boundaries and free surfaces
The frequency factor in the grain-boundaries is less
than in the lattice. D0b < D0l
Energy barrier of diffusion in grain boundaries to be
lower because lower strain of the lattice is needed for
diffusion to take place, the grain boundary possessing
already a substantial level of distortions whose free
energy provide for the activation energy of diffusion.
For this reason the activation energy Q, is less for
grain-boundary diffusion than for volume or lattice
diffusion. Qb < Ql
Q is still lower for surface diffusion and for “pipe”
diffusion along grain boundaries.
This difference in Q values explains why short-circuit
diffusion plays a significant role only at diffusion
temperatures below about three-quarters of the
absolute melting point, Tf °K.
15. Short-circuit paths of diffusion, diffusion along
grain boundaries and free surfaces
Because of the very rapid movements of atoms on free surfaces, surface diffusion plays an important role in a large
number of metallurgical phenomena.
However, grain-boundary diffusion is of more immediate concern because, in the average metallic specimen,
the grain-boundary area is many times larger than the surface area. Furthermore, grain boundaries form a
network that passes through the entire specimen. It is this latter property that often causes large errors to appear
in the measurement of lattice diffusion, coefficients.
When the diffusivity of a metal is measured with polycrystalline samples, the results are usually representative of
the combined effect of volume and grain-boundary diffusion. What is obtained, therefore, is an apparent
diffusivity, Dapp, which may not correspond to either Dl or Db.
However, under certain conditions, Db may be small, so that Dapp = Dl . On the other hand, if the conditions are right,
Db may be so large that Dapp diverges considerably from Dl. Let us investigate these conditions.
15. Short-circuit paths of diffusion, diffusion along
grain boundaries and free surfaces
Diffusion in a polycrystalline specimen cannot be described as a simple summation of diffusion through the
crystals and along the boundaries.
Diffusion in the boundaries tends to progress more rapidly than diffusion through the crystals, but this effect is
counteracted because as the concentration of solute atoms builds up -in the boundaries, a steady loss of
atoms occurs from the boundaries into the metal on either side of the boundary.
The nature of this process can be visualized with the aid of Fig. 12.21, which represents a diffusion couple composed of
two pure metals A and B.
Short arrows represent the nature of the movement of A atoms into the B matrix.
Parallel arrows perpendicular to the weld interface represents the volume
component of diffusion. Arrows parallel to the grain boundaries indicate the
movement of atoms along boundaries, and Arrows perpendicular to the
boundaries represent the diffusion from the boundaries into the crystals.
The smaller the grain size d, the greater the total grain-boundary
area available for boundary diffusion and, therefore, the greater
the importance of boundaries in the diffusion process.
15. Short-circuit paths of diffusion, diffusion along
grain boundaries and free surfaces
The importance of grain-boundarydiffusion phenomena in diffusion
measurements is also a function of
temperature. (Fig. 12.22 12.23)
Single crystal specimens
Polycrystalline specimens
(grain size d = 35 µm before
diffusion anneal)
Lattice or diffusion when compared with
boundary diffusion, is more sensitive to
temperature change. Lattice diffusion
and
Total or apparent diffusivity Dapp
grain-boundary
diffusion
different temperature dependence.
have
15. Short-circuit paths of diffusion, diffusion along
grain boundaries and free surfaces
Only a small fraction of the cross section
Single crystal specimens
Polycrystalline specimens
(grain size d = 35 µm before
diffusion anneal)
of a typical metal is employed by a given
short-circuit
mechanism.
For
grain-
boundary diffusion the fraction is about 10-5 .
Therefore, the ratio Db/Dl must approach 105
before the amount of material transported
along the grain boundaries is comparable to
that diffusing in the volume of the grains.
At high temperatures the ratio is less than this value, but with decrease in temperature Db decreases less than
Dl because of the difference in their Q values.
The apparent diffusion coefficient, Dapp begins to deviate from Dl at a sufficiently low temperature because of the
increasing contribution of grain-boundary diffusion.
At still lower temperatures, where both volume diffusion and grain-boundary diffusion may be ineffective,
significant atomic transport can occur along the "pipe'" of disturbed lattice surrounding a dislocation line.
15. Short-circuit paths of diffusion, diffusion along
grain boundaries and free surfaces
• Thus as the temperature is raised, the rate of diffusion through the lattice increases more rapidly than the rate
of diffusion along the grain boundaries.
• Conversely, as the temperature is lowered, the rate of diffusion along the grain boundaries decreases less
rapidly.
• The net effect is that at very high temperatures diffusion through the lattice tends to overpower the
grain boundary component, but at low temperatures diffusion at the grain-boundary component becomes
more and more important in determining the total or apparent diffusivity Dapp.
•
• Diffusivities determined with polycrystalline specimens are more liable to be representative of lattice diffusion if
they are measured at high temperatures.
• The reliability of the data can be increased by controlling the grain size of the specimens. The larger the grain
size, the smaller the grain-boundary contribution to the diffusivity.
• Thus for accurate measurements of lattice diffusivities, using polycrystalline specimens, high temperatures and
large-grained specimens should be used.
16. Fick’s first law in terms of mobility and an
effective force
16. Fick’s first law in terms of mobility and an
effective force
16. Fick’s first law in terms of mobility and an
effective force
17. Diffusion in non-isomorphic systems
17. Diffusion in non-isomorphic systems
17. Diffusion in non-isomorphic systems
17. Diffusion in non-isomorphic systems
17. Diffusion in non-isomorphic systems
18. Application of diffusion concept to the
homogenization of castings
It is often of interest to be able to calculate the time taken for an inhomogeneous alloy to reach complete
homogeneity, as for example in the elimination of segregation in castings.
18. Application of diffusion concept to the
homogenization of castings
18. Application of diffusion concept to the
homogenization of castings
18. Application of diffusion concept to the
homogenization of castings
Summary
• Make sure you understand language and concepts:
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Diffusion
Driving force of diffusion and concentration gradient
Energy barrier and activation energy
Self-diffusion
Interdiffusion
Diffusion coefficient
Vacancy diffusion
Interstitial diffusion
Flux of diffusion
Rate of diffusion
Fick’s first and second laws
Steady-state diffusion
Nonsteady-state diffusion
Solutions to Fick’s laws of diffusion
Temperature dependence of diffusion coefficient D