Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 3 Part 4 pps

Beke, Dislocation and Grain-boundary Diffusion in metallic Systems, Chap.. Balluffi, Grain-Boundary Diffusion Mechanisms in Metals, in: Diffusion in Crystalline Solids, G.E.. Balluffi, Grain-Bo

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Techno-33 Dislocation Pipe Diffusion

Atomic migration in solids is more rapid along or close to dislocations thanthrough the regular lattice (Fig 31.1) Since practically all crystals containdislocations, any measured diffusion rate will usually contain a dislocationcontribution This may be quite negligible for low dislocation densities, espe-cially at high temperatures However, it can become important at low temper-atures because of the low activation enthalpy for dislocation diffusion relative

to that of lattice diffusion (see Chap 31) It may be even the dominant mode

of transport in some diffusion-controlled processes observable at relativelylow temperatures such as precipitation and metal oxidation The study andunderstanding of dislocation diffusion is therefore a matter of importance

It is common practise to denote the number of dislocations that penetrate

the unit area, ρ d, as the dislocation density It corresponds to the total length

of dislocation lines per unit volume of a crystal For example, a typical cation density of a well-annealed metal is about 106 cm−2 This corresponds

dislo-to a dislocation length of 10 km per cm3 In a heavily deformed metal thedislocation length can reach 105to 106km per cm3

The average distance Λ between dislocations depends on the dislocation

arrangement It is usually given by

Then, the overlap of diffusion fields from neighbouring dislocations is gible Type C kinetics occurs for

negli-a > √

when diffusion is restricted to the dislocation core The latter is characterised

by the dislocation pipe radius a (see below).

The subject of dislocation diffusion has been reviewed by Gibbs andHarris[1], Gjostein [2], and Balluffi [3] A collection of data for diffusionalong dislocations in metals can be found in Le Claire’s chapter in a datacollection for metals [4] and in a chapter by Erdelyi and Beke in a datacollection for non-metals [5] A thorough mathematical analysis analogous tothe analysis of grain-boundary diffusion in Chap 32 has been given by LeClaire and Rabinovitch [6] The main features of their treatment andmajor results are summarised below

33.1 Dislocation Pipe Model

The simplest model for discussing diffusion properties of dislocations has beenintroduced by Smoluchowski [8] and is illustrated in Fig 33.1 Dislocations

are considered as cylindrical pipes of radius a The diffusivity in the cation pipe, D d , is larger than the lattice diffusivity, D, outside the pipe.

dislo-A frequent assumption for the pipe radius is a = 0.5 nm.

More realistic models would take into account that the diffusion coefficient

varies with the distance r from the dislocation core However, there is no clear indication for a suitably simple form of D d (r) Luther [9] investigated the consequences of D d (r) ∝ 1/r2 The advantage of Luther’s approach is notvery apparent Therefore, we prefer to regard dislocation diffusion as beingadequately represented by Smoluchowski’s model This approach is analogous

to the Fisher model of grain-boundary diffusion

Fig 33.1 Smoluchowski model of a dislocation pipe

33.1 Dislocation Pipe Model 585The main features of dislocation diffusion can be illustrated for the case of

isolated dislocations If c and c d represent the concentrations of the diffuseroutside and inside the dislocation pipe, Fick’s equations to be solved are:

∂c

∂t = D

1

2c

∂z2



for r < a (33.5)

In Eqs (33.5) r and z are cylindrical coordinates They denote the distance

from the pipe axis and from the surface, respectively The solutions are

sub-ject to the boundary conditions at r = a There must be continuity of fluxes

For isolated dislocations the additional boundary conditions are

∂c

As in the case of grain-boundary diffusion, two initial conditions at the surface

z = 0 are considered: constant source (case I)

and instantaneous source (case II)

In case I, the concentration at the surface is maintained at a value c0 for all

times t ≥ 0 In case II, a very thin layer of 2M diffuser atoms per unit area

is deposited on the surface

Constant source conditions are appropriate, for example, if diffusion

oc-curs from a vapour phase Then c0is the concentration of the diffuser at thesurface in equilibrium with the vapour Constant source conditions are alsoappropriate for solubility-limited diffusion Instantaneous source conditionssimulate a conventional thin-film tracer diffusion experiment However, it isonly fully appropriate in practice if there is no rapid surface diffusion towards

the dislocation to compensate the loss near r = 0 due to rapid diffusion down

the dislocation pipe In practice, neither the constant source condition northe instantaneous source condition may exactly describe the situation prevail-ing at the surface However, very likely they do represent the limits betweenwhich any experimental condition will lie

While the diffusion fields for the constant source and the instantaneoussource are different [6], we shall see below that the gradient of the log ¯c versus z plot in the dislocation tail is the same for both cases This is analo-

gous to the Whipple and the Suzuoka solutions for grain-boundary diffusion(see Chap 32) For diffusion of a solute that segregates to dislocations with

an equilibrium segregation factor s, the second equation of (33.6) may be replaced by sc = c d (see also Chap 32)

Exact solutions of the problem for isolated dislocations and for a onal array of parallel dislocations all normal to and ending at the surface

hexag-z = 0 of a semi-infinite solid have been worked out by Le Claire and

Ra-binovitch [10–12] and summarised by the same authors [6] For simplicity,

we consider self-diffusion along dislocations, which implies s = 1 The tions for the concentrations field around a dislocation, c(r, z, t), are of the

solu-form

c(r, z, t) = c1 (z, t) + c2(r, z, t) for r ≥ a (33.10)

c1represents the standard expression for the concentration in the absence of

the dislocation, under constant or instantaneous source conditions c2 is theadditional concentration outside dislocations due the rapid diffusion downand out of them The solutions for the diffusion fields are rather difficult tohandle We confine ourselves to the expressions for the average concentration,

¯

c(z, t), in a thin layer at some depth z after some time t.

33.2 Solutions for Mean Thin Layer Concentrations

Mean concentrations are of prime interest for the analysis of dislocation fusion as for grain-boundary diffusion Mean concentrations are measured

dif-in serial sectiondif-ing experiments It is convenient to dif-introduce abbreviationsanalogous to the normalised variables of grain-boundary diffusion:

¯

c I (η) = c0

$erfcη

33.2 Solutions for Mean Thin Layer Concentrations 587For the thin-layer source (upper index II), the solution for the mean concen-tration ¯c II is analogous to the Suzuoka solution of grain-boundary diffusion.

It consits of a thin-film solution plus a dislocation tail:

¯

c II (η) = √ M

πDt

exp

In both cases, the contribution of dislocation diffusion is proportional to the

total cross-sectional area of dislocations πa2ρ d In Eqs (33.12) and (33.13)

the quantities Q I and Q II are given by the following expressions [6]:

θ = 2zY1(zα) + (x2β − z2α)Y0(zα) ,

φ = 2zJ1 (zα) + (x2β − z2α)J0 (zα) , (33.16)

where J1, J0and Y1, Y0 denote Bessel functions of the first and second kind

of order one and zero, respectively We note that constant source and taneous source solutions are related via

instan-¯

c II (η, t) = − M

c0 √ Dt

∂

∂η¯c

Fig 33.2 compares the dependence of Q I and Q II on the normalised depth

variable η Both Q I and Q II first increase as η increases, pass through a imum, and then decrease monotonically Q I is always positive and becomes

max-zero at η = 0; Q II has a zero and changes sign at the η-value for which Q I is

at its maximum, in accordance with Eq (33.17) The value of Q II is negative

for smaller η-values because the finite amount of diffuser available under

in-stantaneous source conditions is depleted around the dislocation pipe by the

rapid diffusion down the pipe At larger η-values, the mean concentration

is enhanced in both cases as diffuser is leaking out of the dislocation pipeinto its surrounding The properties illustrated in Fig 33.2 for dislocationpipe diffusion are qualitatively similar to the corresponding quantities forgrain-boundary diffusion shown in Fig 32.12

When measurements are extended to values of the reduced penetration

depth η such that the first terms of Eqs (33.12) or (33.13) are negligible,

Fig 33.2 Dislocation diffusion: mean thin-layer concentrations of the constant

and instantaneous source solutions, Q I and Q II , for α = 10 −2 and (∆− 1) = 105

according to Le Claire and Rabinovitch [6]

dislocation tails can be observed in penetration profiles The concentration

in the tails is due to the material that has diffused down and out of thedislocations to depths well below those reached by lattice diffusion alone The

tail properties are determined by Q I and Q II Figure 33.3 show logarithmic

plots of Q I and Q II versus η for various values of the parameters α and αβ.

These plots reveal the following features:

Fig 33.3 Dislocation diffusion: constant source solution Q I (left) and neous source solution Q II (right) versus η for α = 10 −1 , 10 −2 , 10 −3 , αβ = 10 (full lines) and αβ = 102 (dashed lines) according to Le Claire and Rabinovitch [6]

instanta-33.2 Solutions for Mean Thin Layer Concentrations 589

1 Beyond η-values of 4 to 5 the plots are practically linear for α ≤ 1 This

implies that plots of log ¯c are linear versus z for dislocation tails We

recall that plots of log ¯c for grain-boundary tails are linear versus z 6/5

2 For given values of α and αβ, the slopes of the linear regions are

prac-tically the same for constant and instantaneous source conditions Theslopes can be represented as

The properties of A(α) are illustrated in Fig 33.4 A(α) is of the order

of unity and a slowly varying function of α with very weak dependence

on αβ.

3 In principle, the dislocation tail can provide an estimate of the dislocationdensity, if the experimental accuracy is sufficient to permit an extrapo-

lation back to z = 0 The intersect with the ordinate, ¯ c int, is a measure

of the dislocation density For details we refer the reader to the review of

Le Claire and Rabinovitch[6]

Fig 33.4 Dislocation diffusion: The quantity A(α) is plotted as a function of α

for various values of αβ according to Le Claire and Rabinovitch [6]

Type B kinetics: The solutions in Eqs (33.12) and (33.13) correspond to

isolated dislocations They are appropriate if the average dislocation distance

Λ is larger than the lattice diffusion length, i.e for Λ  √ Dt Then,

disloca-tions are sufficiently apart from each other and the individual diffusion fields

do not overlap and type B kinetics prevails According to Eq (33.18) themeasured slope of the dislocation tail is, for either surface condition, givenby

∂ ln ¯ c

as long as A(α) is insensitive to variations in αβ The quantity (D d /D − 1)a2

can be determined from the slope in the tail region of a plot of logarithm ¯c versus penetration distance z with little uncertainty, because A(α) depends

only weakly on α In addition, the quantity ∆ ≡ D d /D can be determined

from the slope of the tail, provided that the pipe radius a is known Obviously the result for ∆ depends on a There are no undisputed measurements for a.

A frequent assumption for metals is a = 0.5 nm [13, 14] Larger values prevail

in ionic crystals because of electrostatic effects

Profiles of dislocation diffusion under type B kinetics conditions are ilar to grain-boundary diffusion, with two major differences:

sim-(i) As discussed above, diagrams of log ¯c versus z are linear for dislocation

tails In contrast, grain-boundary tails are linear in plots of logarithm

¯

c versus z 6/5 In practice, a distinction between dislocation and boundary diffusion on the basis of the shape of the penetration plotsmay be difficult It requires measurements over at least two orders ofmagnitude in the tail region

grain-(ii) Because A(α) is only a slowly varying function of α, and thus an even

more slowly varying function with time, the slopes of dislocation tails arealmost independent of time This is in marked contrast to grain-boundarytails, where the slopes in a plot of log ¯c versus z 6/5 are proportional to

t −1/4 (see Chap 32) This permits a relatively sensitive test to

distin-guish dislocation and grain-boundary diffusion tails This test can, forexample, also be used to distinguish diffusion along isolated dislocationsfrom dislocations grouped in low-angle grain boundaries

Type A kinetics: For long diffusion anneals and high dislocation densities

type A kinetics condition prevail This is the case for Λ < √

1 G.B Gibbs, J.E Harris, in: Interfaces Conference, Australian Institute of

Met-als, Butterworth, Melbourne, 1969

2 N.A Gjostein, Chap 9 in: Diffusion, Am Soc for Metals, Metals Park, Ohio,

1973

3 R.W Balluffi, Phys Stat Sol 42, 11 (1970)

4 A.D Le Claire, Diffusion in Dislocations, Chap 11 in: Diffusion in Solid als and Alloys, H Mehrer (Vol Ed.), Landolt-B¨ornstein, Numerical Data andFunctional Relationships in Science and Technology, New Series, Group III:Crystal and Solid State Physics, Vol.26, Springer-Verlag, 1990, p 626

Met-5 G Erdelyi, D.L Beke, Dislocation and Grain-boundary Diffusion in metallic Systems, Chap 11 in: Diffusion in Semiconductors and Non-metallic Systems, D.L Beke (Vol Ed.), Landolt-B¨ornstein, Numerical Data and Func-tional Relationships in Science and Technology, New Series, Group III: Con-densed Matter, Vol.33, Subvolume B1, Springer-Verlag, 1999, p 11–1

Non-6 A.D Le Claire, A Rabinovitch, The Mathematical Analysis of Diffusion in locations, in: Diffusion in Crystalline Solids, G.E Murch, A.S Nowick (Eds.),

Dis-Academic Press, Inc., 1984, p 259

7 L.G Harrison, Trans Faraday Soc 57, 1191 (1961)

8 R Smoluchowski, Phys Rev 87, 482 (1952)

9 L.C Luther, J Chem Phys 43, 2213 (1965)

10 A.D Le Claire, A Rabinovitch, J Phys C: Solid State Physics 14, 3863 (1981)

11 A.D Le Claire, A Rabinovitch, J Phys C: Solid State Physics 15, 3345 (1982),

erratum 5727

12 A.D Le Claire, A Rabinovitch, J Phys C: Solid State Physics 16, 2087 (1983)

13 H Mehrer, M L¨ubbehusen, Defect and Diffusion Forum 66–69, 591 (1989)

14 Y Shima, Y Ishikawa, H Nitta, Y Yamazaki, K Mimura, M Isshiki, Y Iijima,

Materials Transactions, JIM, 43, 173 (2002)

15 E.W Hart, Acta Metall 5, 597 (1957)

‘nanotech-to several thousand a‘nanotech-toms Nanoscience probably first gained attention in

a 1959 lecture of the American Nobel laureate of 1965 in physics RichardFeynman (1918–1988), who stated ‘ that the day was not far off, when

substances could be assembled at an atomic level’ Although this day has not

yet really come, nanotechnology involves at least manufacturing and terisation of materials with crystal grains of nanometer size

charac-In nanocrystalline materials, new atomic structures and properties aregenerated by utilising the atomic arrangement in the cores of defects such

as grain boundaries, interfaces, and dislocations Depending on the type ofdefects utilised, nanocrystalline materials with different structures can begenerated These materials consist of a large volume fraction of defect coresand (strained) crystal lattice regions

As an example, Fig 34.1 shows the structure of a two-dimensionalnanocrystalline material The crystals are represented by periodic arrays

of atoms in different crystallographic orientations (full circles) The atomicstructures of the core regions of the boundaries between the crystallites aredifferent because their structure depends on the crystal misorientations and

on the boundary inclinations The boundary core regions (open circles) arecharacterised by a reduced atomic density and by interatomic spacings devi-ating from those in the crystallites Nanocrystalline materials are sometimesalso denoted as nanophase materials or as nanometer-sized crystalline mate-rials

In this chapter, we consider mainly bulk nanocrystalline materials Ourunderstanding of nanocrystalline materials is documented in reviews, e.g.,

by Siegel and Hahn [1], Birringer and Gleiter [2], Gleiter [3],Gialanella and Lutterotti[4], Heitjans and Indris [6], and Chad-wick[7]

Diffusion has attracted attention, largely because material transport longs to the group of physical properties differing most from single-crystalline