Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 3 Part 3 pot

grain-boundary energy plays a central rˆole in grain-boundary diffusion and insegregation of foreign atoms to the boundary.. If the twin boundary does not lie exactly parallel to the twin

Fig 32.2 Low-angle tilt boundary after Burgers [19]

Low-angle twist boundaries can be considered as a planar network of screwdislocations [23]

As the misorientation Θ between the two grains increases, the strain fields

of dislocations progressively cancel each other so that γ increases slower than

Eq (32.2) predicts In general, when the misorientation exceeds 10 to 15degrees the dislocation spacing is so small that the dislocation cores overlap

It is then impossible to identify individual dislocations At this stage, thegrain-boundary energy is almost independent of misorientation High-angleboundaries contain large areas of poor fit and have a relatively open structure(Fig 32.3) The bonds between the atoms are broken or highly distortedand consequently the grain-boundary energy is relatively high Correlationsbetween the macroscopic parameters of grain boundaries and their energyhave been explored by atomistic computer simulations For a review, we referthe reader to the already mentioned anthology of Wolf and Yip [22] The

Fig 32.3 Random high-angle grain boundary (schematic)

grain-boundary energy plays a central rˆole in grain-boundary diffusion and insegregation of foreign atoms to the boundary As a rule of thumb, high-anglegrain-boundary energies are often found to be about one third of the energy ofthe free surface In low-angle boundaries, however, most of the atoms fit verywell into both lattices so that there is little free volume and the interatomicbonds are only slightly distorted The regions of poor fit are restricted todislocation cores.

32.2.2 Special High-Angle Boundaries

Not all high-angle boundaries have an open disordered structure Specialhigh-angle boundaries have significantly lower energies than random high-angle boundaries Special boundaries occur at particular misorientations ofthe grains and orientations of the boundary plane which allow the adjoin-ing lattices to fit together with relatively little distortion of the interatomicbonds

The simplest special high-angle boundary is the boundary between twins

If the boundary is parallel to the twinning plane, the atoms in the boundary

fit perfectly into both grains The result is a coherent twin boundary trated in Fig 32.4 In fcc metals the twinning plane is a close-packed{111}

illus-plane Twin orientations in fcc metals correspond to a misorientation of 70.2degrees around a

boundary between the twin-related crystals The atoms in such a boundaryare essentially in undistorted positions and the energy of a coherent twinboundary is very low in comparison to the energy of a random high-angleboundary

If the twin boundary does not lie exactly parallel to the twinning plane theatoms do not fit perfectly into each grain and the boundary energy is higher.Such boundaries are denoted as incoherent twin boundaries The energy of

a twin boundary is very sensitive to the orientation of the grain-boundaryplane If the boundary energy is plotted as a function of the boundary orien-tation (right part of Fig 32.4) a sharp cusped minimum is obtained at theposition of the coherent boundary

Low grain-boundary energies are also found for other large-angle aries A two-dimensional example is shown in Fig 32.5 This is a symmetrical

bound-Fig 32.4 A coherent twin boundary (left) Twin-boundary energy γ as a function

of the orientation φ of the grain-boundary plane (right)

Fig 32.5 A special large-angle boundary according to Gleiter [24]

tilt grain boundary between grains with a misorientation of 38.2 degrees Theboundary atoms fit rather well into both grains leaving little free volume.Moreover, a small group of atoms is repeated at regular intervals along theboundary

High-resolution transmission electron microscopy (HREM) can be used toresolve the atomic structure of a grain boundary (see, e.g., [26]) Interfacessuitable for HREM are tilt boundaries, whose tilt axis coincides with a low-index zone axis Figure 32.6 shows the HREM micrograph of a (113) [113]symmetric tilt boundary in a gold bicrystal This grain boundary is periodicand several grain-boundary units along the boundary can be identified This

image also illustrates that the grain-boundary width δ (see below) is of the

order of 0.5 nm

Fig 32.6 A high-resolution transmission electron microscope image of a (113)[113]

symmetric tilt boundary in gold according to Wolf and Merkle [25]

32.3 Diffusion along an Isolated Boundary

(Fisher Model)

Most of the mathematical treatment of grain-boundary diffusion is based onthe model first proposed by Fisher [7] The grain boundary is represented

by a semi-infinite, uniform, and isotropic slab of high diffusivity embedded

in a low-diffusivity isotropic crystal (Fig 32.7) The grain boundary is

de-scribed by two physical parameters: the grain-boundary width δ and the grain-boundary diffusivity D gb The latter of course depends on the grain-boundary structure discussed above It is usually much larger than the lattice

diffusivity D in the adjoining grains, i.e D gb  D The grain-boundary width

is of the order of an interatomic distance δ ≈ 0.5 nm is a widely accepted

value (see above)

In a tracer diffusion experiment a layer of tracer atoms (either self- orforeign atoms) is deposited at the surface Then, the specimen is annealed at

constant temperature T for some time t During the annealing treatment the

labeled atoms diffuse into the specimen in two ways:

(i) by lattice diffusion directly into the grains and

(ii) much faster along the grain boundary

Atoms which diffuse along the grain boundary eventually leave it and continuetheir diffusion path in the grains, thus giving rise to a lattice diffusion zonearound the grain boundary The total concentration of the diffuser in the

specimen is the result of two contributions: a concentration c, established

Fig 32.7 Fisher’s model of an isolated grain boundary D: lattice diffusivity,

Dgb : diffusivity in the grain boundary, δ: grain-boundary width

either directly by in-diffusion from the source or by leaking out from the

grain boundary and the concentration inside the grain boundary, c gb.Mathematically, this diffusion problem can be described by applyingFick’s second law to diffusion inside the grains and inside the grain-boundaryslab For composition-independent diffusivities we have:

In Eqs (32.3) the coordinate system was chosen in such a way that the

xz-plane is the symmetry plane of the grain boundary Then, the tion field depends on the variables y and z Continuity of the concentrations

concentra-and of the diffusion fluxes across the interfaces between grain-boundary concentra-andgrains require the following boundary conditions:

c( ±δ/2, z, t) = c gb(±δ/2, z, t) (32.4)and

Since the grain-boundary width is very small (δ ≈ 0.5 nm) and D gb  D,

one can simplify the problem (see, e.g., [8]) and arrive at the following set oftwo coupled equations:

lat-It is convenient to introduce normalised variables, which correspond to

the spatial coordinates y, z and to time t, respectively:

is a dimensionless parameter, which equals the ratio of grain-boundary and

lattice diffusivity In physical terms, the variable ξ accounts for the extent of

lateral lattice diffusion from the grain boundary into the grains The quantity

η accounts for the influence of direct lattice diffusion from the source into the grains; the smaller η the stronger is this influence Whereas the physical meaning of ξ and η are obvious this is according to the author’s experience rather less so for the parameter β, also called the Le Claire parameter It is

a measure of the extent to which grain-boundary diffusion is enhanced relative

to lattice diffusion Loosely speaking, one can consider β as the ratio of the transport capacity inside the grain-boundary slab, c gb D gb δ, to the transport capacity along the grain-boundary fringe, cD √

Dt, which has a width √

Dt.

As we shall see below, diffusion profiles in bi- or polycrystals usuallyconsist of a near-surface part dominated by lattice diffusion and a deep pen-etrating grain-boundary tail Grain-boundary tails of the concentration field

tend to level out as β decreases Then, it becomes more difficult to reveal

the influence of enhanced diffusion along grain boundaries in experiments

A question in this context is, what are the optimum conditions for the

deter-mination of grain-boundary diffusivities? The quantity β is relevant for this

question This can be seen from Fig 32.8, in which isoconcentration contours

are plotted for various values of β The dotted line corresponds to the ing case, D gb = D, for which preferential grain-boundary diffusion is absent.

limit-The isoconcentration contours illustrate that the penetration of the diffuseralong the grain boundary is much greater than anywhere else in the crystal

The larger the value of β, the more pronounced is the lateral diffusion fringe along the grain boundary For an accurate determination of D gb from section-

ing experiments (see below) β must be at least 10 The annealing conditions

must be chosen accordingly

The solution for diffusion along an isolated grain-boundary slab embedded

in a crystal can be written as follows:

c(ξ, η, β) = c1(η) + c2(ξ, η, β) (32.10)

in the grains and

Fig 32.8 Isoconcentration contours for various values of the Le Claire

parameter β

inside the boundary In Eq (32.10) the first term represents in-diffusion intothe grains from the external source The second term represents the leaking-out contribution from the grain boundary The direct grain-boundary con-

tribution, c gb, can be neglected when √

Dt  δ; studies of the direct

grain-boundary diffusion require √

Dt < δ These distinctions are also related to

the kinetic regimes B and C of diffusion in polycrystals, discussed later inthis chapter

Constant Source Solution: Let us at first consider the case of a constant

source (also called infinite or inexhaustible source), with the diffuser tration kept constant at the surface and zero everywhere inside the sample

concen-at the beginning The initial and boundary conditions are:

c(y, 0, t) = c0 for t > 0 ,

c(y, ∞, 0) = 0

c o is the concentration of the diffuser at the surface in the source

An approximate solution of the diffusion problem formulated in Eqs (32.6),(32.7), and (32.12) was given already by Fisher [7] An exact solution hasbeen worked out three years later by Whipple [27] using the Fourier-Laplacetransformation method (see Chap 3) We shall not go through the long andrather tedious mathematical exercise of deriving it A transparent derivation

of this solution can be found, e.g., in a textbook by Adda and ert [28] The first term of Eq (32.10) is a complementary error function

and represents direct in-diffusion into the grains from the inexhaustiblesource The second term in Eq (32.10) represents the leakage contributionfrom the grain boundary into the grains It is given by



dσ ,

(32.14)

where σ is an integration variable Note that the time variable is included in

η and also in the Le Claire parameter β At a fixed temperature β ∝ 1/ √ t, i.e β decreases with increasing time (see Eqs 32.8).

Instantaneous Source (or Thin-Film) Solution: For an instantaneous

source initial and boundary conditions are expressed by:

c(y, z, 0) = M δ(z), c(y, z, 0) = 0 for z > 0, c(y, ∞, t) = 0,

∂c(y, z, t)

δ(z) is the Dirac delta function and M the amount of diffuser deposited per

unit area This surface condition entails that the initial layer is completelyconsumed during the diffusion experiment

An exact solution of the diffusion problem formulated in Eqs (32.6),(32.7), and (32.15) has been worked out by Suzuoka [29, 30], using themethod of Fourier-Laplace transforms (see Chap 3) The first term in

σ 3/2

erfc

12



A comparison between the constant source solution, Eq (32.14), and theinstantaneous source solution, Eq (32.17), shows that the latter can be ob-tained from Eq (32.14) by a transformation through the operator− √ Dt ∂/∂η

Fig 32.9 Concentration contours for constant source (left) and a thin-film source

solutions (right) for an arbitrary value of β = 50 according to Suzuoka [30]

and replacing c0 by M Furthermore, in the case of the instantaneous source

solution it can be shown that

∞

0

A consequence of this equation is that the total amount of diffuser is given

by the volume diffusion term c1(η), thus establishing that the total amount

M of diffuser is conserved.

Figure 32.9 shows a comparison between the two types of diffusion sources

for an arbitrarily chosen value of β = 50 For the thin-layer source the boundary term c2 is negative near the surface, indicating that in the near-surface region the crystal is supplying diffusing material to the grain bound-ary The reason is that the source concentration decreases much more rapidly

grain-at the grain boundary than anywhere else Thus, in the near-surface regionthe grain boundary behaves as a ‘sink for the diffuser’ Beyond a certaindepth the grain-boundary behaviour changes to that of a ‘source for the dif-fuser’, since then the direct volume diffusion from the source is negligible

In contrast, for an inexhaustible source the contribution c2 is always itive This implies that the grain boundary behaves as source of diffuser,irrespective of whether the near-surface region or the deeper regions are con-sidered

pos-Average Concentrations in Thin Layers: pos-Average concentrations are of

prime interest for the analysis of grain-boundary diffusion experiments, which

are carried out by the radiotracer technique (see Chap 13) Let us thereforediscuss an expression for the average concentration in a thin layer, ¯c, at some depth z (Fig 32.7) The total amount in a thin section between z − ∆z/2 and z + ∆z/2 and parallel to the free surface is given by the integral

[c(y, z, t) + c gb (y, z, t)] dxdydz (32.19)

L x and L are the dimensions of the bicrystal along the x − and y−axes, spectively The quantity L x L∆z is the section volume For sake of simplicity,

re-let us assume that the grain boundary lies in the center of the bicrystal and

that the section is so thin that the concentration along the z −axis remains

constant within a section Then, the average concentration ¯c is obtained by

¯

c = 1L

We know from Eq (32.10) that the concentration field has two contributions,

where c1represents bulk diffusion and c2is the grain-boundary leakage bution given either by the thin-film solution (32.17) or by the constant-source

contri-solution (32.14) Since c1 is constant in the xy-plane, the bulk diffusion tribution to the average concentration equals c1 and we have

3 The same assumption is made in the next section for to type B kinetics in

polycrystals In type C kinetics the direct grain-boundary contributions is inating

dom-In terms of the dimensionless variables ξ, η, β, and ∆, we get

¯

c(η, β) = c1(η, t) +2

√ Dt L

√

π − Y erfcY



dσ , (32.26)where

Y = σ − 1 2β

2η

√ π

per unit area of the bicrystal An expression which is generally valid for

a bicrystal or for polycrystals is:

In Eq (32.29) λ represents the grain-boundary length per unit area on the

sample surface exposed to the diffuser

For a bicrystal with a width L normal to the grain-boundary, Λ and λ are

For the more complicated case of a polycrystal with random distribution

of grain size the reader may consult the textbook of Kaur, Mishin, andGust[8]

Segregation of Foreign Atoms: Foreign atoms tend to segregate into

grain boundaries This process is called grain-boundary segregation and sults in an excess concentration of the foreign element in the grain bound-ary The mathematics of grain-boundary diffusion discussed so far has nottaken into account segregation effects This is justified for grain-boundaryself-diffusion

re-Diffusion of foreign elements (solutes) can be treated by the same matics, if grain-boundary segregation is taken into account in a suitable way

mathe-In the case of self-diffusion, it was assumed that the grain-boundary width δ is

a purely geometric quantity and that the matching condition Eq (32.4) ply expresses the continuity of concentration across the grain/grain-boundaryinterfaces This assumption must be modified for solute atoms because theycan segregate into the boundary Accoording to Gibbs [31] segregation can

sim-be taken into account by introducing the segregation factor s The matching

condition then reads

sc( ±δ/2, z, t) = c gb(±δ/2, z, t) (32.33)Equation (32.33) rests on two assumptions:

1 The solute atoms in the grain boundary maintain local thermodynamicequilibrium with the solute atoms in the lattice adjacent to the interfaces

In other words, segregation is in local equilibrium at any depth z.

2 The grain-boundary segregation follows the law of Henry, which reads

c gb = sc Henry’s law implies that the segregation factor is a function of temperature only and not a function of c Henry’s law is applicable when both, c gb and c, are small enough This is usually the case, when impurity

diffusion in a pure matrix is studied in radiotracer experiments