COMPUTATIONAL MATERIALS ENGINEERING Episode 8 doc

Equation 5.48 shows that the macroscopic diffusion coefficient D, from its purelyphenomenological definition of Fick’s first law, is directly related to the microscopic mobil-ityB , which

y x

FIGURE 5-6 Mass conservation in diffusion.

On the other hand, the concentrationcof atoms in the control volume is

Equation (5.24) is known as Fick’s second law It is a second-order linear partial differentialequation describing transient diffusion processes, and it is interesting that Fick’s second lawfollows directly from Fick’s first law and mass conservation considerations.

From this general form of the diffusion equation, Fick’s second law can also be written inplanar, cylindrical, and spherical coordinates by substituting for the Laplace operator∇2 Usingthe symbolsx,y, andzfor cartesian coordinates,r,θ, andzfor cylindrical coordinates, andr,

θ, andφfor spherical coordinates, the diffusion equation reads

of the linear diffusion equations then becomes

(5.31)

which is an ordinary parabolic differential equation in the independent variableλ The indexp

has the values 1, 2, and 3 for planar, cylindrical, and spherical geometry For the case ofp= 1,

this equation is also known as the Boltzmann di ffusion equation.

5.3.2 Solutions to Fick’s Second Law

In this section, we will briefly review some important solutions to Fick’s second law [equation(5.24)]

Spreading of a Diffusant from a Point Source

Consider one-dimensional diffusion of a massM along thex-direction in an infinite sampleunder the assumption that the entire mass is initially concentrated in a single plane According

to the law of mass conservation, we have

 +∞

The solution to equation (5.24) with the constraint (5.32) is closely related to the problem ofrandom walk of atoms and it is elaborated in many textbooks (e.g., ref [Gli00]) Accordingly,for unit amount of mass, we have

c (x, t) = M

2√ πDtexp

−x24Dt



(5.33)

Equation (5.33) shows that, if atoms spread according to Fick’s second law, they will form

a Gaussian distribution (see Figure 5-7) The general solution to a unit mass swarm of atomsspreading into an infinite sample of dimensiondis

where r is the distance from the point source Equation (5.34) represents the probability

distribution for a diffusant spreading into an infinite sample of dimensiondfrom a general pointsource This equation has also been used in the derivation of Einstein’s equation inSection 5.2.3

Diffusion into a Semi-infinite Sample

Consider one-dimensional diffusion of a diffusant into a semi-infinite sample This is a differentsituation now compared to the spreading diffusant from before, since we assume a continuoussupply of atoms entering the sample from the outside boundary This type of boundary condition

is typical for surface treatments, such as carburization Again, we start from Fick’s second law.The present boundary conditions are now a constant concentration of atoms at the semi-infiniteboundary, that is,

x (Length)

0 0.5 1.0 1.5 2.0

0.025

0.05 0.1 0.25 1

The solution for this diffusion problem is

c (x, t) = c0· erfc

x

√ 4Dt

4Dtis plotted on thex-axis [compare equation (5.36)]

5.3.3 Diffusion Forces and Atomic Mobility

If a forceFacts on a body, according to Newtonian mechanics, this force accelerates the bodyalong the direction of the force Typical examples of such forces are, for instance, gravitationacting on a mass or an electric field acting on a charged particle Generally, a force can bewritten as the gradient of a potentialΦand we have

against movement in terms of friction with the coefficient of frictiong f, we find the effectivedrift velocityuof a large number of atoms under the forceFfrom the force balance and we canwrite

and with the definition of the activity coefficienta = fX[compare equation (2.67)] we obtain

∇µ = RT (∇ ln f + ∇ ln X) (5.44)Equation (5.44) can be rearranged as

∇µ = RT X

Equation (5.48) shows that the macroscopic diffusion coefficient D, from its purely

phenomenological definition of Fick’s first law, is directly related to the microscopic

mobil-ityB , which has been defined based on atomistic considerations, by a thermal factor RT and

a chemical contribution, which describes the deviation from solution ideality in form of the

logarithmic derivative of the activity coefficientf

An interesting consequence of the preceding considerations is that, even in the absence ofchemical interactions between the species, atoms experience a force which causes a net flux

of atoms Since this force cannot be measured directly, it is often considered as a generalized

force In an ideal solution, the generalized forceFis directly proportional to the concentrationgradient In nonideal solutions, the chemical contribution can be included in the thermodynamicfactorφ, which is

X i= 1[equation (2.22)] must be taken into account and the variation∂X iof

one element can only be performed against the equivalent variation−∂Xrefof some referenceelement

Consider a binary system A–B From equation (5.42), the flux of atoms A is

The choice of reference component in equation (5.56) is somewhat arbitrary and needsfurther attention If we consider the chemical contribution to the diffusion coefficient in a dilutealloy, it is reasonable to use a single reference component since the majority of all exchangeprocesses will include this one single component In concentrated alloys, however, the flux ofcomponent A in the laboratory frame of reference (see Section 5.3.4) is compensated by a flux

of atoms of different kinds depending on the alloy composition In this case, a weighted meanvalue instead of a single reference component is more appropriate, and the thermodynamic fac-torφcan then be expressed as

individ-a single-component system is denoted individ-as the self-di ffusion coefficient D ∗ The self-diffusioncoefficient gives a measure for the effective displacement of atoms caused by random vacancymovement

The radioactive isotope technique can likewise be used to measure “self-diffusion” cients in solutions of multiple components Accordingly, the specimen is prepared as a chem-ically homogeneous solution of given composition and some of the atoms are replaced byradioactive isotopes The diffusion coefficient measured by this method is commonly denoted

coeffi-as the tracer or impurity diffusion coefficient The same symbol D ∗is commonly used for this

quantity and, due to the absence of chemical interactions, the general relation between atomicmobilityBandD ∗holds6

When finally looking at the thermodynamic factor in the asymptotic limit of dilute solution,

we have to recall the analysis of solution thermodynamics in the dilute solution limit (Section2.2.5) With the help of the regular solution model, we have found that the activity coefficientf

approaches a constant value if the solute content goes to zero (Henry’s law) Consequently, thelogarithmic derivative of the activity coefficient in the definition of the thermodynamic factor

φin equation (5.50) becomes zero and the thermodynamic factor thus approaches unity Theimportance of the thermodynamic factor comes into play only in concentrated alloys, whereφ

accounts for the influence of solution nonideality

6When measuring the diffusion coefficient based on tracer elements, this value is always smaller than the true

self-di ffusion coefficient, which is defined on basis of random vacancy-atom exchanges This effect is known

as the correlation e ffect The factor relating correlated and uncorrelated jumps is a constant for each type of

crystal lattice and it is always less than unity (e.g.,fbcc= 0.727 and ffcc= 0.781) In this book, we will not

further distinguish between the two and assume that the correlation e ffect is implicitly taken into account.

5.3.4 Interdiffusion and the Kirkendall Effect

In Section 5.3.3, the net motion of atoms that is caused by a generalized diffusion force has beendiscussed The chemical potential has been identified as the source of this driving force, and thethermodynamic factor has been introduced to account for nonideal thermodynamic behavior Inthis section diffusion will be analyzed in situations where diffusion of multiple atomic speciesoccurs simultaneously

In the 1940s, a severe controversy about the mechanism of diffusion in crystalline solids wasgoing on One group of scientists promoted the traditional view of diffusion, which assumedthat the diffusional transport of atoms occurs on basis of an atom by atom exchange mechanism.Thereby, one atom exchanges place with another atom based on direct exchange(see Figure 5-1) or the ring-exchange mechanism, which involves four atoms that rotate simul-taneously and thus change place The second group of scientists believed that diffusion occurs

by the vacancy-exchange mechanism, that is, atoms switch place only with a neighboring emptylattice site (vacancy), and the transport of atoms occurs as a consequence of the random walk

of these vacancies In fact, the type of diffusion mechanism has substantial influence on therate of diffusion of individual atomic species Consequently, by careful analysis of appropriate

diffusion experiments, it should be possible to identify the predominating diffusion mechanism

in solid-state matter These experiments and the corresponding theoretical analysis will now bediscussed

Consider a binary diffusion couple with the pure substances A and B After bringing A and

B into contact, the A atoms will spread into the B-rich side and vice versa If diffusion occurs

by direct atomic exchange, the macroscopically observed diffusivities of the two atomic speciesmust be identical, because one single exchange process moves the same amount of A and Batoms and diffusion of A and B occurs at the same rate In contrast, if diffusion is carried byatom/vacancy exchange, the A and B atoms can move independently and the diffusivity of thetwo species can be different

Moreover, if one species diffuses from the left to the right side of the diffusion couple byvacancy/atom exchange, the flux of atoms must be accompanied by a counterflux of vacancies

If the diffusivities of A and B are identical, the two vacancy fluxes balance and annihilate In thecase where the diffusivities differ, a net flux of vacancies must occur, which “blows” through

the sample The net flux of vacancies is commonly known as vacancy wind As a result of

the vacancy wind, the lattice of the sample moves in the parallel direction of the vacancy flux

If, by some experimental technique, individual lattice planes, for example, the initial contactarea of the diffusion couple, are marked, movement of the lattice planes can be recorded as afunction of the difference of the diffusive fluxes of the A and B atoms These experiments havebeen carried out by Ernest Kirkendall (1914–2005), an American metallurgist, between 1939and 1947

In a series of three papers [KTU39, Kir42, SK47], Kirkendall and co-workers investigatedthe diffusion of copper and zinc in brass In the experiments of the third paper, the initial contactplane of a diffusion couple between pure copper and brass with 70wt% copper/30wt% zinc hasbeen marked with thin molybdenum wires, such that the movement of the inert markers can

be observed at different stages of the experiment Kirkendall showed that the markers moved

relative to the laboratory frame of reference, which is a reference frame that is fixed to the

sample surrounding An experimentalist looking at a diffusion couple will observe the diffusionprocess in the laboratory frame of reference

From the results of the experiments, Kirkendall drew the following two conclusions:

1 the diffusion of zinc is much faster than the diffusion of copper, and

2 the movement of the markers is related to the difference in the diffusion coefficients

The results of his experiments have been published against the strong resistance of individualresearchers, in particular Robert Franklin Mehl, an American metallurgist (1898–1976) Nowa-

days, the observed Kirkendall drift of the marker plane is considered to be the first striking

proof of the predominance of the vacancy exchange mechanism over direct atomic exchange in

diffusion

A short time after publication of Kirkendall’s papers, L Darken [Dar48] published the firstquantitative analysis of Kirkendall’s experiments, which will briefly be outlined later Consider

the two intrinsic fluxes JA and JB (i.e., the fluxes that are observed when looking at di

ffu-sion from a frame of reference that is fixed to an individual lattice plane, the lattice frame of

reference) according to Figure 5-9 In the steady state case, using Fick’s first law, we have

JA= −DA∂cA

∂r

JB= −DB∂cB

The net flux of atoms across this lattice planeJnetis given as the sum of the intrinsic fluxes

of the components and we obtain

JVa(A)

JVa(B) JA

JB

FIGURE 5-9 Schematic of the Kirkendall effect: Atoms A (gray circles) diffuse into the B-rich side at

a higher rate than the B atoms (white circles) into the A-rich side The net flux of atoms Jnetcauses

a shift of the position of the initial contact plane with a velocity v relative to the laboratory frame of reference, that is, the fixed corners of the specimen The movement of the marker plane is parallel

to the net flux of vacancies J Va,net

Equation (5.60) is known as Darken’s first equation From mass conservation, it is apparent

that a net fluxJnetof atoms causes accumulation of matter on one side of the marker plane

If mass is conserved, this accumulation must be compensated by a shift of the marker planeinto the opposite direction With the mole fractionX i = c i Vmand with the assumption that thepartial molar volumesV i = Vmof each element are identical, the velocity of the marker plane

addition of a convective term or drift term with the convective flux Jconv = vc This type of

transformation is known as Galilean transformation In the laboratory frame of reference, the

flux for species A can be written as

Equation (5.64) is known as Darken’s second equation It is interesting to note that through

an analysis of the concentration profiles in a diffusion couple experiment, only the interdiffusioncoefficientD˜ can be observed To determine the individual intrinsic diffusion coefficientsDAandDB, additional information is necessary, which is exactly the velocity or displacement ofthe marker plane

The interdi ffusion coefficient D˜describes the diffusive fluxes in the laboratory frame of

refer-ence, a convenient frame of reference for human experimentalists The intrinsic diffusion

coef-ficientsDAandDBdescribe the fluxes in the lattice frame of reference, a convenient measurewhen operating on the atomic scale

5.3.5 Multicomponent Diffusion

The concepts that have been introduced in the previous section for binary diffusion are nowgeneralized to multicomponent systems The diffusive flux of atoms has already been defined inSection 5.3.3 and for a componentiwe have

This general form relates the diffusive flux to a mobilityBand a generalized force∂µ/∂r.Since the chemical potentialµ = µ(X)is a function of the local chemical compositionX, equa-tion (5.65) already contains all multicomponent influences that come from solution nonideality

In many practical applications it is convenient to express the diffusive flux in terms of thelocal concentrationscor mole fractionsXinstead of the potential gradient∂µ/∂r In this case,the flux of componentican be obtained by chain derivation of equation (5.65) with

In equation (5.70), the summation is performed over the composition gradients∂c j /∂rof all

nelements, thus summing up the influence of each atomic speciesjon the diffusion behavior

of elementi

However, not all of the fluxesJ i are independent, which is easily demonstrated Since the

sum over allnconcentrationsc iequals the molar density1/Vm(Vmis the molar volume),

Consequently, the concentration gradient of one component can always be expressed in terms

of the gradients of the other ones When choosing component n as the dependent element,