Kinetics of Materials - R. Balluff_ S. Allen_ W. Carter (Wiley_ 2005) Episode 6 pot

Because diffusion along dislocations and crystal sur- faces is comparatively less well characterized, particular attention is paid to grain- boundary transport in this chapter.. 9.1; sec

206 CHAPTER 8 DIFFUSION IN CRYSTALS

The hysteresis loop will therefore appear as a line o f negligible width and slope

1/SR as in Fig 8.24a Negligible internal friction therefore occurs

8.22 Describe in detail how to determine the diffusivity of C in b.c.c Fe using a torsion pendulum Include all of the necessary equations

See Section 8.3.1 and Fig 8.8, where C atoms in sites 1, 2, and 3 expand

the crystal preferentially along z, y, and z , respectively

Solution Using a torsion pendulum, find the anelastic relaxation time, T , by measuring the frequency o f the Debye peak, u p , and applying the relation W ~ T = 1 Having T , the relationship between T and the C atom jump frequency r is found by using the procedure

t o find this relationship for the split-dumbbell interstitial point defects in Exercise 8.5

Assume the stress cycle shown in Fig 8.16 and consider the anelastic relaxation that occurs just after the stress is removed A C atom in a type 1 site can jump into two possible nearest-neighbor type 2 sites or two possible type 3 sites Therefore,

EXERCISES 207

which may be integrated t o obtain

8.23

[ ci(t) - - c;] = [ ~ ( 0 ) - - c;] = e -6r’t (8.178) The relaxation time is then T = 1/(6l?), and because the total jump frequency is

r = 4l?, T = 2/(3r) According t o Eq 7.52, D Z = r r 2 / 6 because f = 1, and because

r = a / 2 , DI = ra2/24 Substituting for r,

(8.179) Finally, insert the experimentally determined value o f T into Eq 8.179 t o obtain DI

Under equilibrium conditions in a stressed b.c.c Fe crystal, interstitial C atoms are generally unequally distributed among the three types of sites iden- tified in Fig 8.8b This occurs because the C atoms in sites 1, 2, and 3 in

Fig 8.8b expand the crystal preferentially along the 2, y, and z directions, respectively These directions are oriented differently in the stress field, and the C atoms in the various types of sites therefore have different interaction energies with the stress field In the absence of applied stress, this effect does

not exist and all sites are populated equally In Exercise 8.22 it was shown

that when the stress on an equilibrated specimen is suddenly released, the re- laxation time for the nonuniformly distributed C atoms to achieve a random distribution, T , is T = 2/(3r), where r is the total jump frequency of a C

atom in the unstressed crystal

Show that when stress is suddenly applied to an unstressed crystal, the relax- ation time for the randomly distributed C atoms to assume the nonrandom distribution characteristic of the stressed state is again T = 2/(3r)

Assume the energy-level system for the specimen shown in Fig 8.25

Write the kinetic equations for the rates of change of the concentrations

of the interstitials in the various types of sites and solve them subject

to the appropriate initial and final conditions Assume that the barri-

ers to the jumping interstitials shown in Fig 8.25 are distorted by the differences in the site energies (indicated in Fig 8.21)

Figure 8.25:

atom in sites 1, 2, or 3 illustrated in Fig 8.8

Energy-level diagram for a stressed b.c.c specimen containing an interstitial

Solution Let c1, c2, and c3 be the concentrations of interstitials occupying sites of types 1, 2, and 3, respectively Also, c1 +c2 +c3 = ctot = constant Since an interstitial

208 CHAPTER 8: DIFFUSION IN CRYSTALS

in a given type of site can jump into two sites of each other type,

of stress, and expanding Boltzmann factors of the form exp[-U,,,/(kT)] t o first order

c1 (m) = ci‘ c2(m) = c;q (8.182)

where el( ) and c2(m) are the final equilibrium concentrations reached at long times

in the presence of the applied stress In view of the symmetry of Eqs 8.180, we try

Finally, the equilibrium concentrations obtained in Eqs 8.184 from the kinetic equations agree with those obtained using equilibrium statistical mechanics In the three-level system in Fig 8.25, the occupation probability for level 1 is

Since c1 = ctotpl, the result for CI is the same as that given by Eq 8.184 Similar agreement is obtained for c2

Rapid diffusion along line and planar crystal imperfections occurs in a thin region centered on the defect core For a dislocation, the region is cylindrical, roughly two interatomic distances in diameter, and includes the “bad material” in the dislocation core.’ For a grain boundary, the region is a thin slab, roughly two interatomic distances thick, including the bad material in the grain boundary core For a free surface, this region is the first few atomic layers of the material at the surface These regions are very thin in comparison to the usual diffusional transport distances To model the diffusion due to these imperfections, we replace them by thin slabs or cylinders of effective thickness, 6, possessing effective diffusivities which are much larger than the diffusivity in the adjoining crystalline material Table 9.1 lists the

Bad material is disordered material in which the regular atomic structure characteristic of the crystalline state no longer exists Good bulk material is free of line or planar imperfections

Kinetics of Materials By Robert W Balluffi, Samuel M Allen, and W Craig Carter 209

Copyright @ 2005 John Wiley & Sons, Inc

210 CHAPTER 9 DIFFUSION ALONG CRYSTAL IMPERFECTIONS

Table 9.1: Notation for Short-circuit Diffusivities

D D (undissoc) diffusivity along an undissociated dislocation core (i.e., a cylin-

der, or a “pipe” of diameter, 6)

D D (dissoc) diffusivity along a dissociated dislocation core (i.e., a cylinder,

or a “pipe” of diameter, 6)

DB diffusivity along a grain boundary (i.e., a slab of thickness, 6)

DS

D X L

diffusivity along a free surface (i.e., a slab of thickness, 6)

diffusivity in a bulk crystal free of line or planar imperfections

D L diffusivity in a liquid

notation to be used to describe the diffusivities in various regions of crystalline materials containing line and planar imperfections

Figure 9.1 presents self-diffusivity data for *DD (dissoc), *DD (undissoc), *DB,

*Ds, *DxL, and *DL, for f.c.c metals on a single Arrhenius plot With the excep- tion of the surface diffusion data, the data are represented by ideal straight-line Arrhenius plots, which would be realistic if the various activation energies were constants (independent of temperature) However, the data are not sufficiently accurate or extensive to rule out some possible curvature, at least for the grain

boundary and dislocation curves, as discussed in Section 9.2.3

Dislocations, grain boundaries, and surfaces can possess widely differing struc- tures, and these structural variations affect their diffusivities to significant degrees

If the defective core region is less dense or “looser” than defect-free material, or

if a defect possesses structurally “open” channels in its core structure, transport will generally be more rapid along the defect, particularly in the open directions Some grain boundary structures can be represented by dislocation arrays, and their boundary diffusivity can be modeled in terms of transport along the grain-boundary

9.1: THE DIFFUSION SPECTRUM 211

dislocation cores General grain boundary structures cannot support discrete local- ized dislocations but, nevertheless, still act as short-circuit diffusion paths

Short-circuit diffusion along grain boundaries has been studied extensively via experiments and modeling Because diffusion along dislocations and crystal sur- faces is comparatively less well characterized, particular attention is paid to grain- boundary transport in this chapter However, briefer discussions of diffusion along dislocations and free surfaces are also presented

To describe the effects of grain-boundary structure on boundary diffusion, it is necessary to review briefly some important aspects of boundary structure Addi- tional details appear in Appendix B It takes a minimum of five geometric pa- rameters to define a crystalline interface Three describe the crystal/crystal mis- orientation: e.g., two to specify the axis about which one crystal is rotated with respect to the other, and one for the rotation angle The remaining two parameters define the inclination of the plane along which the crystals abut at the interface.2

If the interface is a free surface, just two parameters are required to specify the surface's inclination (unit normal) Crystal symmetries determine special values of the parameters at which the interfacial energies take on extreme values Depending

on the specific nature of a system with interfaces, some of the parameters may be constrained and others free to vary as the system seeks a lower-energy state

Small-angle grain boundaries have crystal misorientations less than about 15"

and consist of regular arrays of discrete dislocations (Le., where the cores are sep- arated by regions of defect-free material) As the crystal misorientation across the

boundary increases beyond about 15", the dislocation spacing becomes so small

that the cores overlap and the boundary becomes a continuous slab of bad mate- rial; these are called large-angle boundaries Large-angle boundaries can be further classified into singular boundaries, vicinal boundaries, and general b ~ u n d a r i e s ~

An interface is regarded as singular with respect to a degree of freedom if it is

at a local minimum in energy with respect to changes in that degree of freedom It

is therefore stable against changes in that degree of freedom

A vicinal interface is an interface that deviates from being singular by a rela- tively small variation of one or more of its geometric parameters from their singular- interface values A vicinal interface can therefore minimize its energy by adopting

a fit-misfit structure consisting of patches of the nearby minimum-energy singular interface delineated by arrays of discrete interfacial dislocations or steps as illus- trated in Figs B.4 and B.9 These line defects serve to accommodate the relatively small deviations of the vicinal interfaces from the singular interfaces

A general interface is not energy-minimized with respect to any of its degrees

of freedom, and is far from any singular-interface values of the parameters that set its degrees of freedom Such an interface cannot reduce its energy by adopting a fit-misfit structure (as in the vicinal case) and therefore cannot support localized dislocations or steps Two examples serve to clarify these distinctions:

Example 1 The tilt grain boundary in Fig B.4a is singular with respect to its

tilt angle.4 The boundary in Fig B.4c is vicinal to the singular boundary

2Additional variables may be required, such as three that specify a relative translation of one crystal with respect to the other

3Similar terminology is used for classification of free-surface structure

4See Appendix B for descriptions of tilt, twist, and mixed grain boundaries

212 CHAPTER 9: DIFFUSION ALONG CRYSTAL IMPERFECTIONS

with respect to its tilt angle It consists of patches of the singular boundary delineated by dislocations that accommodate the change in tilt angle

Example 2 A surface corresponding to the patch of light-colored atoms in Fig B.l

is singular with respect to its inclination about an axis parallel to the surface steps in the figure The stepped surface in Fig B.l is vicinal to such a flat surface and consists of patches of the flat surface delineated by steps that accommodate the change in surface inclination

Because the structure of general large-angle grain boundaries is usually less reg- ular and rigid than that of singular or vicinal boundaries, its activation energies for diffusion are typically lower and the diffusivities correspondingly higher The diffusion rate along small-angle grain boundaries is generally lower than along large- angle grain boundaries and, indeed, approaches D x L as the crystal misorientation approaches zero This is due to two factors: first, the diffusion rate along the bad material in dislocation cores is about the same as, or lower than, that along large-angle grain boundary cores (see Fig 9.1); second, because small-angle grain boundaries consist of periodic arrays of lattice dislocations at discrete spacings that approach infinity as the crystal misorientation approaches zero, the density of fast- diffusion paths is smaller in small-angle boundaries than in large-angle boundaries Figure 9.2 presents diffusivity data for a series of tilt boundaries as a function

of the misorientation tilt angle

The structures of these boundaries vary considerably as the misorientation changes

In the central part of the plot, the minima occur at crystal misorientations (values of

Q) corresponding to singular and vicinal boundaries The ends of the plot (where the crystal misorientation approaches zero) correspond to small-angle boundaries, and the diffusivities are correspondingly low The regions centered around the maxima

in Fig 9.2 correspond to general grain boundaries Polycrystalline materials not subjected to special processing conditions possess mainly general boundaries; the grain-boundary data in Fig 9.1 are for general boundaries that have fairly similar diffusivities and can therefore be described reasonably well by average normalized values

9.1: THE DIFFUSION SPECTRUM 213

The wide range of diffusivity magnitudes evident in the diffusivity spectrum

in Fig 9.1 may be expected intuitively; as the atomic environment for jumping becomes progressively less free, the jump rates, r, decrease accordingly in the sequence rS > rB x rD(undissoc) > rD(dissoc) > rXL The activation energies for these diffusion processes consistently follow the reverse behavior,

E S < E B FZ ED(undissoc) < ED(dissoc) < E x L (9.1)

The diffusivity in free surfaces is larger than that in general grain boundaries, which

is about the same as that in undissociated dislocations Furthermore, the diffusivity

in undissociated dislocations is greater than that in dissociated dislocations, which

is greater than that in the c r y ~ t a l : ~

*Ds > *DB M *DD(undissoc) > *DD(dissoc) > *DxL ( 9 4

Free-surface and grain-boundary diffusivities in metals at 0.5Tm are seven to eight orders of magnitude larger than crystal diffusivities Provided that defects are present at sufficiently high densities, significant amounts of mass transport can occur in crystals at 0.5Tm via surface and grain-boundary diffusion even though the cross-sectional area through which the diffusional flux occurs is relatively very small

As the temperature is lowered further, the ratio of diffusivities becomes larger and short-circuit diffusion assumes even greater importance Generally, similar behavior

is found in ionically bonded crystals, as shown in Fig 9.3

There are many situations, particularly at low temperatures, where short-circuit diffusion along grain boundaries and free surfaces is the dominant mode of diffu- sional transport and therefore controls important kinetic phenomena in materials;

5We discuss diffusion along dislocations and free surfaces in Sections 9.3 and 9.4

214 CHAPTER 9: DIFFUSION ALONG CRYSTAL IMPERFECTIONS

several examples are discussed in Sections 9.2 and 9.4 Similar conclusions hold for dislocation diffusional short-circuiting, although to a lesser degree because of the relatively small cross sections of the high-diffusivity pipes

9.2 DIFFUSION ALONG GRAIN BOUNDARIES

9.2.1

In a polycrystal containing a network of grain boundaries, atoms may migrate in both the grain interiors and the grain boundary slabs [4] They may jump into or out

of boundaries during the time available, and spend various lengths of time jumping

in the grains and along the boundaries Widely different situations may occur, depending upon such variables as the grain size, the temperature, the diffusion time, and whether the boundary network is stationary or moving For example,

as the grain size is reduced and more boundaries become available, the overall diffusion will be enhanced due to the relatively fast diffusion along the boundaries

At elevated temperatures where the ratio of the boundary diffusivity to the crystal diffusivity is lower than at low temperatures (Fig 9.1), the importance of the boundary diffusion will be diminished At very long diffusion times, the distance each atom diffuses will be relatively large, and each atom will be able to sample a number of grains and grain boundaries If the boundaries are moving, an atom in

a grain may be overrun by a moving boundary and be able to diffuse rapidly in the boundary before being deposited back into crystalline material behind the moving boundary

Consider first the relatively simple case where the boundaries are stationary and each diffusing atom is able to diffuse both in the grains and along at least several grain boundaries during the diffusion time available This will occur whenever the diffusion distance in the grains during the diffusion time t is significantly larger than the grain size [i.e., approximately when the condition *DXLt > s2 (where s is the grain size) is satisfied] For each atom, the fraction of time spent diffusing in grain boundaries is then equal to the ratio of the number of atomic sites that exist

in the grain boundaries over the total number of atomic sites in the specimen [5] This fraction is q x 36/s: for each atom, the mean-square displacement due to diffusion along grain boundaries is then *DBqt, and the mean-square displacement

in the grains is *DxL(l - q)t The total mean-square displacement is then the sum

of these quantities, which can be written

Regimes of Grain-Boundary Short-circuit Diffusion in a Polycrystal

considerably longer than the average grain size (b) Regime B: the diffusion length in the grains is significant but smaller than the grain size ( c ) Regime C: the diffusion length in the grains is negligible, but significant diffusion occurs along the grain boundaries In all figures, preferential penetration within the grain boundaries is too narrow to be depicted

overlaps multiple boundaries Note that in Fig 9.4a, fast grain-boundary diffusion will cause preferential diffusion to occur along the narrow grain-boundary cores beyond the main diffusion front, but the number of atoms will be relatively small and this effect cannot be depicted

900°C 700°C 500°C 350°C

1 .o 1.2 1.4 1.6 lOOO/T (K-I)

Figure 9.5: Values of the average self-diffusivity (*D) in single- and polycrystalline silver At lower temperatures pain-boundary diffusion makes significant contributions to the overall measured average di usivity in the polycrystal From Turnbull [7]

216 CHAPTER 9 DIFFUSION ALONG CRYSTAL IMPERFECTIONS

At the opposite extreme when essentially no diffusion occurs in the grains but significant diffusion still occurs along the boundaries, the overall diffusion will con- sist of only diffusion penetration along the boundaries, as illustrated in Fig 9 4 ~ This will tend to occur at low temperatures or short times under the conditions

*DXLt < X2 and *DBt > X2, where X is the interatomic distance

Many intermediate cases may also occur in which diffusion takes place in both the boundaries and in the grains but where the diffusion length in the grains is smaller than the grain size, as in Fig 9.4b The conditions for this type of dif- fusion are X2 <* DXLt < s2 and *DBt > X2 The latter two regimes are called

isolated-boundary diffusion regimes, since in both cases there is no overlap of the diffusion fields associated with the individual boundary segments, as in the multiple-

boundary regime The three types of regimes just described are often termed the

A, B, and C regimes, as indicated in Fig 9.4, corresponding to Harrison’s original designation [6]

When the boundaries move during the diffusion, as they might during grain growth or recrystallization, the situation is considerably altered If w is the aver- age boundary velocity, the boundaries will be essentially stationary when w t < A, and the regimes described above will again pertain However, when the condi- tion [ v ‘ m + wt] > s is satisfied, the multiple-boundary diffusion regime will

hold, and Eq 9.4 will apply even if is negligible, since in such a case the boundaries visit the atoms rather than vice versa Conversely, when the condition

[m + wt] < s is satisfied, the isolated-boundary diffusion regime will exist The various regimes of possible diffusion behavior can be represented graphically

in an approximate manner, as shown in Fig 9.6 [8] The axes are taken to be log(*DXLt) and log(wt): logarithmic scales have been used to show the details near the origin because s/X is typically lo3 or more The stationary-boundary regimes

Crystal diffusion ahead of grain

into adjacent grains S I N X L E Stationary boundaries, Isolated boundary diffusion and N o crystal (XL) diffusion penetration into adjacent grains M I + X L G Moving boundaries, Isolated boundary diffusion CI ystal ( X L ) diffusion ahead of boundaries M.1

NXL E Moving boundaries Isolated boundary diffusion, N o crystal ( X L ) diffusion ahead

of boundaries S 0 M M = Stationary O r Moving boundaries, Multiple boundary diffusion

From Cahn and Balluffi [8]

9.2: GRAIN BOUNDARIES 217

(vt < A) are shown on the left and include Harrison's A, B, and C regimes The isolated-boundary regimes are enclosed in a region that includes the origin and extends out along the vertical and horizontal axes to distances where *DXLt = s2

and vt = s, respectively Beyond the isolated-boundary regimes the multiple- boundary regime holds sway in all locations

The isolated-boundary regime for moving boundaries in Fig 9.6 is subdivided into two regimes, depending on whether the crystal diffusion is fast enough so that the atoms are able to diffuse out into the grains ahead of the advancing boundaries

To analyze this, consider a boundary segment between two grains moving with velocity u as in Fig 9 7 ~

Atoms are diffusing into the boundary laterally from its edges and can diffuse out through its front face into the forward grain At the same time, atoms will be deposited in the backward grain in the wake of the boundary In the quasi-steady state in a coordinate system fixed to the moving boundary, the diffusion flux in the forward grain is J = -*DXL(dc/dx) - wc and the diffusion equation is

with the solution

dc

dx

* D ~ ~ - + wc = A

where A is a constant At a large distance in front of the boundary, dcldx + 0 and

c + 0 and therefore A = 0 Finally, upon integration,

(9.7)

= c~ e - v x / * D x L

where cG is the concentration maintained at the boundary The resulting concen- tration profile is shown in Fig 9.7b According to Eq 9.7, the concentration in front of the boundary will be negligible when *DxL/v < A Therefore, the curve separating the regimes indicated by M.I.XL and M.I.NXL in Fig 9.6 should follow the straight-line relationship *DXLt = Xvt, as indicated

The diagram in Fig 9.6 is highly approximate, but it is useful for visualizing the various regimes that might be expected during diffusion in a polycrystal With increasing time, the point representing the system will start at the origin and move

in the grain behind the boundary Diffusion is in a steady state in a coordinate system moving with the boundary (b) Tracer concentration in the vicinity of the boundary according to

Eq 9.7

218 CHAPTER 9 DIFFUSION ALONG CRYSTAL IMPERFECTIONS

progressively away from it If diffused long enough it will inevitably reach the multiple-boundary regime, regardless of whether the boundaries are stationary or moving

9.2.2

A Regime: Since diffusion in this regime is macroscopically similar to diffusion in

a homogeneous material possessing an effective bulk diffusivity (Eq 9.4) it may be analyzed by the methods described in Chapters 4 and 5

B Regime: In this regime, the diffusant diffuses along a boundary while simulta-

neously leaking out by diffusing into the adjoining grains Analysis of this type of diffusion is therefore considerably more complex than for diffusion in the A and C regimes since it involves solving for the coupled diffusion fields in the grain bound- ary and in the adjoining grains This problem has been solved to different degrees

of accuracy for several boundary conditions [9] Solutions are generally obtained

that contain the lumped grain-boundary diffusion parameter p = 6*DB and the crystal diffusivity *DxL The analyses can then be applied to experimental results

to obtain values of p when *DxL is known

Fisher has produced a relatively simple solution for a specimen geometry that

is convenient for experimentalists and which has been widely used in the study of boundary self-diffusion by making several approximations which are justified over

a range of conditions [9, lo] The geometry is shown in Fig 9.8; it is assumed that the specimen is semi-infinite in the y direction and that the boundary is station- ary The boundary condition at the surface corresponds to constant unit tracer concentration, and the initial condition specifies zero tracer concentration within the specimen Rapid diffusion then occurs down the boundary slab along y while tracer atoms simultaneously leak into the grains transversely along z by means of crystal diffusion The diffusion equation in the boundary slab then has the form

Analysis of Diffusion in the A, B, and C Regimes

distance 1c1, from the boundary at scaled depth, y l , from the surface Penetration distance

in grains is assumed large relative to 6

9 z GRAIN BOUNDARIES 219

where 5 1 , y1, and t l are reduced dimensionless variables defined by x1 = 216,

y1 = ( y / 6 ) J w , and t l = t*DXL/d2 In the process of solving this equa- tion, Fisher found that the combination of relatively fast diffusion along the grain boundary and slower leakage into the grains causes the concentration in the bound- ary slab to quickly “saturate” so that the concentration in the adjoining grains

at the time t l is essentially the same as the concentration that would have been

there if the concentration distribution which existed in the boundary at t l had been maintained constant there since the start of the diffusion at t l = 0 Also, since the diffusion along the boundary is rapid compared to the transverse diffusion in the grains, gradients along y in the grains are much smaller than gradients along x

in the grains and can therefore be neglected The transverse concentration profile along X I in the grains at constant y1 is therefore, to a good approximation, an error-function type of solution (Eq 5.23) of the form

Also, the rapid saturation found in the boundary slab produces a quasi-steady-state condition along the boundary: dcB/dtl in Eq 9.8 can then be set equal to zero so that

(9.10) d2CB(Yl, tl )

The results above and the approximations made to obtain them have been shown

to be valid when the dimensionless parameter p = d*DB(*DXL)-3/2t-1/2 2 20 [9] When this condition is not satisfied, more rigorous but complex analysis is required Experimentalists have frequently used Eq 9.12 to determine values of the lumped grain boundary diffusion parameter p = 6*DB The specimen is diffused for the time

t and is sectioned by removing thin slices parallel to the surface of thickness Ay The tracer content of each slice, AN, is then measured and plotted logarithmically against y From Eq 9.12 the resulting curve should have the slope

and p = 6*DB can then be determined when *DxL is known

Further analyses of B-regime diffusion, including diffusion under different bound- ary conditions, are described by Kaur and Gust [9]

When solute atoms rather than tracer isotope atoms diffuse in the B regime, further analysis is necessary Solute atoms may be expected to segregate in the

220 CHAPTER 9: DIFFUSION ALONG CRYSTAL IMPERFECTIONS

grain boundary, and the concentration in the boundary slab cf(y1,tl) will then differ from the concentration in the grains in the direct vicinity of the boundary slab cFL(0, y1, tl) Assuming local equilibrium between these concentrations and that

a simple McLean-type segregation isotherm typical of a dilute solution applies [4], the two concentrations will be related by

vs yl but it now contains the segregation ratio Ic Values of SDf can therefore be obtained only when independent information about k is available Further analysis

is required if the simple McLean isotherm does not apply and k is concentration- dependent

C Regime: In this regime, diffusion occurs only in the thin grain-boundary slabs

Since the number of diffusing atoms within the slabs is exceedingly small, the experimental measurement of boundary concentration profiles is difficult Recourse has therefore been made to accumulation methods where the number of atoms which have diffused along a grain boundary are collected in a form that can readily be measured For example, solute atoms have been deposited on one surface of a thin- film specimen possessing a columnar grain structure and then diffused through the film along the grain boundaries so that they accumulated on the reverse surface [ll,

121 The diffusion was carried out at low temperatures where no crystal diffusion occurred, and where, according to Fig 9.1, the diffusion along the surfaces was much more rapid than the diffusion along the grain boundaries Diffusion through the film specimen was therefore controlled by the rate of grain-boundary diffusion Measurement of the rate of accumulation of the solute on the reverse surface then allowed the measurement of the lumped parameter SO? as detailed in Exercise 9.4

9.2: GRAIN BOUNDARIES 221 9.2.3

The mechanisms by which fast grain-boundary diffusion occurs are not well estab- lished at present There is extensive evidence that a net diffusional transport of atoms can be induced along grain boundaries, ruling out the ring mechanism and implicating defect-mediated mechanisms as responsible for grain-boundary diffu- sion [13] Due to the small amount of material present in the grain boundary, it has not been possible, so far, to gain critical information about defect-mediated processes using experimental techniques Recourse has been made to computer simulations which indicate that vacancy and interstitial point defects can exist in the boundary core as localized bona fide point defects (see the review by Sutton and Balluffi [4]) Calculations also show that their formation and migration energies are often lower than in the bulk crystal Figure 9.9 shows the calculated trajectory of

a vacancy in the core of a large-angle tilt grain boundary in b.c.c Fe Calculations showed that vacancies were more numerous and jump faster in the grain boundary than in the crystal, indicating a vacancy mechanism for diffusion in this particular boundary However, there is an infinite number of different types of boundaries, and computer simulations for other types of boundaries indicate that the dominant mechanism in some cases may involve interstitial defects [4, 121

During defect-mediated grain-boundary diffusion, an atom diffusing in the core will move between the various types of sites in the core Because various types of jumps have different activation energies, the overall diffusion rate is not controlled

by a single activation energy Arrhenius plots for grain-boundary diffusion therefore should exhibit at least some curvature However, when the available data are of only moderate accuracy and exist over only limited temperature ranges, such curvature may be difficult to detect This has been the case so far with grain-boundary diffusion data, and the straight-line representation of the data in the Arrhenius

Mechanism of Fast Grain-Boundary Diffusion

Boundary midplane

[ooi]

Figure 9.9: Calculated atom jumps in the core of a C5 symmetric (001) tilt boundary in b.c.c Fe A pair-potential-molecular-dynamics model was employed For purposes of clarity the scales used in the figure are [I301 : [310] : [OOT] = 1 : 1 : 5 All jumps occurred in the fast-diffusing core region Along the bottom, a vacancy was inserted at B and subse uently executed the series of jumps shown The tra'ectory was essentially parallel to the t j t axis Near the center of the figure, an atom in a b site jumped into an interstitial site at I At the top an atom jumped between B , I and B' sites From Balluffi et al [14]

222 CHAPTER 9 DIFFUSION ALONG CRYSTAL IMPERFECTIONS

plot in Fig 9.3 must be regarded as an approximation that yields an effective activation energy, E B , for the temperature range of the data Some evidence for curvature of Arrhenius plots for grain-boundary diffusion has been reviewed [4]

9.3 DIFFUSION ALONG DISLOCATIONS

As with grain boundaries, dislocation-diffusion rates vary with dislocation struc- ture, and there is some evidence that the rate is larger along a dislocation in the edge orientation than in the screw orientation [15] In general, dislocations in close- packed metals relax by dissociating into partial dislocations connected by ribbons

of stacking fault as in Fig 9.10 [16] The degree of dissociation is controlled by the stacking fault energy Dislocations in A1 are essentially nondissociated because

of its high stacking fault energy, whereas dislocations in Ag are highly dissociated because of its low stacking fault energy The data in Fig 9.1 (averaged over the available dislocation orientations) indicate that the diffusion rate along dislocations

in f.c.c metals decreases as the degree of dislocation dissociation into partial dislo- cations increases This effect of dissociation on the diffusion rate may be expected because the core material in the more relaxed partial dislocations is not as strongly perturbed and “loosened up’’ for fast diffusion, as in perfect dislocations

In Fig 9.1, *DD for nondissociated dislocations is practically equal to *DB, which indicates that the diffusion processes in nondissociated dislocation cores and large- angle grain boundaries are probably quite similar Evidence for this conclusion also comes from the observation that dislocations can support a net diffusional transport

of atoms due to self-diffusion [15] As with grain boundaries, this supports a defect- mediated mechanism

The overall self-diffusion in a dislocated crystal containing dislocations through- out its volume can be classified into the same general types of regimes as for a polycrystal containing grain boundaries (see Section 9.2.1) Again, the diffusion may be multiple or isolated, with or without diffusion in the lattice, and the dis- locations may be stationary or moving However, the critical parameters include

*DD rather than *DB and the dislocation density rather than the grain size The multiple-diffusion regime for a dislocated crystal is analyzed in Exercise 9.1 Figure 9.11 shows a typical diffusion penetration curve for tracer self-diffusion into a dislocated single crystal from an instantaneous plane source at the sur- face [17] In the region near the surface, diffusion through the crystal directly from the surface source is dominant However, at depths beyond the range at

,Stacking fault

ribbon

Partial f 2 Partial dislocation 1 dislocation 2

Figure 9.10:

partial dislocations separated by a ribbon of stacking fault

Dissociated lattice dislocation in f.c.c metal The structure consists of two

9.4 FREE SURFACES 223

Dislocation pipe diffusion

C

e

Penetration depth -w

Figure 9.11: Typical penetration curve for tracer self-diffusion from a free surface at

tracer concentration csurf into a single crystal containing dislocations Transport near the

surface is dominated by diffusion in the bulk; at greater depths, dislocation pipe diffusion is the major transport path

which atoms can be delivered by crystal diffusion alone, long penetrating “tails” are present, due to fast diffusion down dislocations with some concurrent spreading into the adjacent lattice and no overlap of the diffusion fields of adjacent dislo- cations This behavior corresponds to the dislocation version of the B regime in Fig 9.4

9.4 DIFFUSION ALONG FREE SURFACES

The general macroscopic features of fast diffusion along free surfaces have many

of the same features as diffusion along grain boundaries because the fast-diffusion path is again a thin slab of high diffusivity, and a diffusing species can diffuse in both the surface slab and the crystal and enter or leave either region For example,

if a given species is diffusing rapidly along the surface, it may leak into the adjoining crystal just as during type-B kinetics for diffusion along grain boundaries In fact, the mathematical treatments of this phenomenon in the two cases are similar The structure of crystalline surfaces is described briefly in Sections 9.1 and 12.2.1 and in Appendix B All surfaces have a tendency to undergo a “roughening” tran- sition at elevated temperatures and so become general Even though a considerable effort has been made, many aspects of the atomistic details of surface diffusion are still unknowns6

For singular and vicinal surfaces at relatively low temperatures, surface-defect- mediated mechanisms involving single jumps of adatoms and surface vacancies are

p r e d ~ m i n a n t ~ Calculations indicate that the formation energies of these defects are of roughly comparable magnitude and depend upon the surface inclination [i.e.,

(hkl)] Energies of migration on the surface have also been calculated, and in most cases, the adatom moves with more difficulty Also, as might be expected, the diffusion on most surfaces is anisotropic because of their low two-dimensional symmetry When the surface structure consists of parallel rows of closely spaced atoms, separated by somewhat larger inter-row distances, diffusion is usually easier parallel to the dense rows than across them In some cases, it appears that the

6 0 u r discussion follows reviews by of Shewmon [18] and Bocquet et al [19]

7Adatoms, surface vacancies, and other features of surface structure are depicted in Fig 12.1

224 CHAPTER 9: DIFFUSION ALONG CRYSTAL IMPERFECTIONS

transverse diffusion occurs by a replacement mechanism in which an atom lying between dense rows diffuses across a row by replacing an atom in the row and pushing the displaced atom into the next valley between dense rows Repetition of this process results in a mechanism that resembles the bulk interstitialcy mechanism described in Section 8.1.3 In addition, for vicinal surfaces, diffusion rates along and over ledges differs from those in the nearby singular regions

At more elevated temperatures, the diffusion mechanisms become more complex and jumps to more distant sites occur, as do collective jumps via multiple defects

At still higher temperatures, adatoms apparently become delocalized and spend significant fractions of their time in “flight” rather than in normal localized states

In many cases, the Arrhenius plot becomes curved at these temperatures (as in Fig 9.1), due to the onset of these new mechanisms Also, the diffusion becomes more isotropic and less dependent on the surface orientation

The mechanisms above allow rapid diffusional transport of atoms along the sur- face We discuss the role of surface diffusion in the morphological evolution of

surfaces and pores during sintering in Chapters 14 and 16, respectively

Bibliography

1 N.A Gjostein Short circuit diffusion In Diffusion, pages 241-274 American Society for Metals, Metals Park, OH, 1973

2 I Herbeuval and M Biscondi Diffusion of zinc in grains of symmetric flexion of

aluminum Can Metall Quart., 13(1):171-175, 1974

Diffusion in ceramics In R.W Cahn, P Haasen, and E Kramer, editors, Materials Science and Technology-A Comprehensive Treatment, volume 11,

pages 295-337, Wienheim, Germany, 1994 VCH Publishers

4 A.P Sutton and R.W Balluffi Interfaces in Crystalline Materials Oxford University Press, Oxford, 1996

5 E.W Hart On the role of dislocations in bulk diffusion Acta Metall., 5(10):597,

8 J.W Cahn and R.W Balluffi Diffusional mass-transport in polycrystals containing stationary or migrating grain boundaries Scripta Metall Mater., 13(6):499-502, 1979

9 I Kaur and W Gust Fundamentals of Grain and Interphase Boundary Diffusion Ziegler Press, Stuttgart, 1989

10 J.C Fisher Calculation of diffusion penetration curves for surface and grain boundary diffusion J Appl Phys., 22(1):74-77, 1951

11 J.C.M Hwang and R.W Balluffi Measurement of grain-boundary diffusion at low- temperatures by the surface accumulation method 1 Method and analysis J Appl

12 Q Ma and R.W Balluffi Diffusion along [OOl] tilt boundaries in the Au/Ag system

1 Experimental results Acta Metall., 41(1):133-141, 1993

13 R.W Balluffi Grain boundary diffusion mechanisms in metals In G.E Murch and

A S Nowick, editors, Diffusion in Crystalline Solids, pages 319-377, Orlando, FL,

1984 Academic Press

3 A Atkinson

Phys., 50(3):1339-1348, 1979

EXERCISES 225

14 R.W Balluffi, T Kwok, P.D Bristowe, A Brokman, P.S Ho, and S Yip Deter- mination of the vacancy mechanism for grain-boundary self-diffusion by computer simulation Scripta Metall Mater., 15(8):951-956, 1981

On measurements of self diffusion rates along dislocations in f.c.c metals Phys Status Solidi, 42(1):11-34, 1970

16 R.E Reed-Hill and R Abbaschian Physical Metallurgy Principles PWS-Kent, Boston, 1992

17 Y.K Ho and P.L Pratt Dislocation pipe diffusion in sodium chloride crystals Radiat

18 P Shewmon Diffusion in Solids The Minerals, Metals and Materials Society, War- rendale, PA, 1989

19 J.L Bocquet, G Brebec, and Y Limoge Diffusion in metals and alloys In R.W Cahn and P Haasen, editors, Physical Metallurgy, pages 535-668 North-Holland, Amsterdam, 2nd edition, 1996

Derive an equation similar to Eq 9.4 for the effective bulk self-diffusivity, (*D),

in the presence of fast dislocation diffusion Assume that the dislocations are present at a density, p, corresponding to the dislocation line length in a unit volume of material

Solution During self-diffusion, the fraction of the time that a diffusing atom spends

in dislocation cores is equal t o the fraction o f all available sites that are located in

the dislocation cores This fraction will be 7 = p7d2/4 The mean-square displace- ment due t o self-diffusion along the dislocations is then *DDqt, while the corresponding displacement in the crystal is *DxL(l - 7)t Therefore,

(*D)t = * D X L ( l - 7)t + *DD7t (9.17) and because 7 << 1,

cf/cf" = k = constant, where cf is the solute concentration in the disloca-

tion cores and cfL is the solute concentration in the crystal

Solution Because the fraction of solute sites in the dislocations is small, the number

of occupied solute-atom sites (per unit volume) in the crystal is c g L , and the number of

226 CHAPTER 9: DIFFUSION ALONG CRYSTAL IMPERFECTIONS

occupied sites in the dislocations is pd2kc?XL/4 The fraction of time that a diffusing solute atom spends in dislocation cores is then 17 = p7d2k/4 Therefore, following the same argument as in Exercise 9.1,

(9.21) for solute diffusion

Solution As indicated in the text, Eq 9.9 must have the form of Eq 9.15 in order

t o satisfy the segregation condition k = cf/c?” at the boundary slab Equation 9.10

%ource” surface to the “accumulation” surface is controlled by the diffusion rate along the transverse boundaries If the diffusant, designated component

2, is initially present on the source surface and absent on the accumulation surface and the specimen is isothermally diffused, a quasi-steady rate of ac- cumulation of the diffusant is observed on the accumulation surface after a short initial transient Derive a relationship between the rate of accumulation

EXERCISES 227

and the parameter SDF that can be used to determine SDf experimentally Assume that each grain is a square of side d in the plane of the surface

c Source surface Fil

thi

Accumulation surface

Figure 9.12:

diffusion

Transport of diffusant through a thin polycrystalline film by grain-boundary

Solution Because of the fast surface diffusion, the concentrations of the diffusant

on both surfaces are essentially uniform over their areas After the initial transient, the quasi-steady rate (per unit area of surface) a t which the diffusant diffuses along the transverse boundaries between the two surfaces is

Here, d is the average grain size of the columnar grains, JB is the diffusional flux along the grain boundaries, dcB/dx = [cB(0) - cB(I)] / I , where cB(0) and cB(I) are the diffusant concentrations in the boundaries at the source surface and accumulation surface, respectively, and I is the specimen thickness In the early stages, c B ( I ) = 0

and, therefore, t o a good approximation,

Solution Equation 9.18 may be solved for p in the form

(9.28)

It is estimated from Fig 9.1 that *DD(dissoc)/*DXL = 3 x lo6 at Tm/T = 2.0 Also,

6 % 6 x lo-* cm-* Using these values and (*D)/*DxL = 2 in Eq 9.28,

p E 10' cmP2 Therefore, it appears that the dislocations could make a significant contribution t o diffusion under many common conditions