Stability of defects

3.7 Stability of defects

3.7.1 Dislocation loops

During annealing, defects such as dislocation loops, stacking-fault tetrahedra and voids may shrink in size. This may be strikingly demonstrated by observing a heated specimen in the microscope. On heating, the dislocation loops and voids act as vacancy sources and shrink. This process occurs in the temperature range where self-diffusion is rapid, and confirms that the removal of the residual resistivity associated with Stage II is due to the dispersal of the loops, voids, etc.

The driving force for the emission of vacancies from a vacancy defect arises in the case of (1) a prismatic loop from the line tension of the dislocation, (2) a Frank loop from the force due to the stacking fault on the dislocation line, since in intermediate and high γ-metals this force far outweighs the line tension contribution, and (3) a void from the surface energy γ s . The annealing of Frank loops and voids in quenched aluminum is shown in Figures 3.52 and 3.54, respectively. In a thin metal foil the rate of annealing is generally controlled by the rate of diffusion of vacancies away from the defect to any nearby sinks, usually the foil surfaces, rather than the emission of vacancies at the defect itself.

To derive the rate equation governing the annealing, the vacancy concentration at the surface of the defect is used as one boundary condition of a diffusion-controlled problem and the second boundary condition is obtained by assuming that the surfaces of a thin foil act as ideal sinks for vacancies. The rate then depends on the vacancy concentration gradient developed between the defect, where the vacancy concentration is given by

c =c 0 exp {(dF/dn)/kT }, (3.14) with (dF/dn) the change of free energy of the defect configuration per vacancy emitted at the

temperature T , and the foil surface where the concentration is the equilibrium value c 0 . For a single, intrinsically faulted circular dislocation loop of radius r the total energy of the defect

F is given by the sum of the line energy and the fault energy, i.e. F∼

2 = 2πr{[μb 2 / 4π(1 − ν)] ln (r/r 0 ) } + πr γ.

In the case of a large loop (r > 50 nm) in a material of intermediate or high stacking-fault energy (γ ˜> 60 mJ m −2 ) the term involving the dislocation line energy is negligible compared with the stacking-fault energy term and thus, since (dF/dn)

= (dF/dr) × (dr/dn), is given simply by γB 2 , where B 2 is the cross-sectional area of a vacancy in the (1 1 1) plane. For large loops the diffusion

148 Physical Metallurgy and Advanced Materials

(a)

(b)

(d) Figure 3.52 Climb of faulted loops in aluminum at 140 ◦

(c)

C. (a) t = 0 min, ( b) t = 12 min, (c) t = 24 min, (d) t = 30 min (after Dobson, Goodhew and Smallman, 1967; courtesy of Taylor & Francis).

geometry approximates to cylindrical diffusion 11 and a solution of the time-independent diffusion equation gives for the annealing rate:

dr/dt = −[2πD/b ln (L/b)][exp(γB 2 /kT ) − 1]

(3.15) = const.[exp(γB /kT ) − 1],

where D =D 0 exp ( −U D /kT ) is the coefficient of self-diffusion and L is half the foil thickness. The annealing rate of a prismatic dislocation loop can be similarly determined; in this case dF/dr is determined solely by the line energy, and then

dr/dt = −[2πD/b ln (L/b)](αb/r) (3.16) = const.[αb/r],

where the term containing the dislocation line energy can be approximated to αb/r. The annealing of Frank loops obeys the linear relation given by equation (3.15) at large r (Figure 3.53); at small r the curve deviates from linearity because the line tension term can no longer be neglected and also

11 For spherical diffusion geometry the pre-exponential constant is D/b.

Crystal defects 149

50 Radius (nm)

0 50 100 Time (s)

Figure 3.53 Variation of loop radius with time of annealing for Frank dislocations in Al showing the deviation from linearity at small r.

because the diffusion geometry changes from cylindrical to spherical symmetry. The annealing of prismatic loops is much slower, because only the line tension term is involved, and obeys an r 2 versus t relationship. In principle, equation (3.15) affords a direct determination of the stacking fault energy γ by substitution, but since U D is usually much bigger than γB 2 this method is unduly sensitive to small errors in U D . This difficulty may be eliminated, however, by a comparative method in which the annealing rate of a faulted loop is compared to that of a prismatic one at the same tempera- ture. The intrinsic stacking-fault energy of aluminum has been shown to be 135 mJ m −2 by this technique.

In addition to prismatic and single-faulted (Frank) dislocation loops, double-faulted loops have also been annealed in a number of quenched fcc metals. It is observed that, on annealing, the intrinsic loop first shrinks until it meets the inner, extrinsically faulted region, following which the two loops shrink together as one extrinsically faulted loop. The rate of annealing of this extrinsic fault may be derived in a way similar to equation (3.15) and is given by

dr/dt = −[πD/b ln (L/b)][exp(γ E B 2 /kT ) − 1]

= const.{exp (γ E B 2 /

2kT ) − 1},

from which the extrinsic stacking-fault energy may be determined. Generally, γ E is about 10–30% higher in value than the intrinsic energy γ. Loop growth can occur when the direction of the vacancy flux is towards the loop rather than away from it, as in the case of loop shrinkage. This condition can arise when the foil surface becomes

a vacancy source, as, for example, during the growth of a surface oxide film. Loop growth is thus commonly found in Zn, Mg, Cd, although loop shrinkage is occasionally observed, presumably due to the formation of local cracks in the oxide film, at which vacancies can be annihilated. Figure 3.45 shows loops growing in Mg as a result of the vacancy supersaturation produced by oxidation. For the double loops, it is observed that a stacking fault is created by vacancy absorption at the growing outer perimeter of the loop and is destroyed at the growing inner perfect loop. The perfect regions expand faster than the outer stacking fault, since the addition of a vacancy to the inner loop decreases

the energy of the defect by γB 2 , whereas the addition of a vacancy to the outer loop increases the energy by the same amount. This effect is further enhanced as the two loops approach each other

150 Physical Metallurgy and Advanced Materials due to vacancy transfer from the outer to inner loops. Eventually the two loops coalesce to give a

perfect prismatic loop of Burgers vector c = [0 0 0 1], which continues to grow under the vacancy supersaturation. The outer loop growth rate is thus given by

˙r 0 = [2πD/B ln(L/b)][(c s / c 0 )

− exp(γB 2 /kT )],

(3.18) when the vacancy supersaturation term (c s / c 0 ) is larger than the elastic force term tending to shrink

the loop. The inner loop growth rate is

(3.19) where exp(

˙r 2 i = −[2πD/B ln(L/b)][(c s / c 0 ) − exp(−γB /kT )],

−γB 2 /kT ) << 1, and the resultant prismatic loop growth rate is

˙r p = −[πD/B ln(L/b)]{(c s / c 0 ) − [(αb/r) + 1]}

(3.20) where (αb/r) < 1 and can be neglected. By measuring these three growth rates, values for γ, (c s / c 0 )

and D may be determined; Mg has been shown to have γ = 125 mJ m −2 from such measurements.

3.7.2 Voids

Voids will sinter on annealing at a temperature where self-diffusion is appreciable. The driving force for sintering arises from the reduction in surface energy as the emission of vacancies takes place from the void surface. In a thin metal foil the rate of annealing is generally controlled by the rate of diffusion of vacancies away from the defect to any nearby sinks, usually the foil surfaces.

The rate then depends on the vacancy concentration gradient developed between the defect (where the vacancy concentration is given by

c =c 0 exp {(dF/dn)/kT }, (3.21) with (dF/dn) the change in free energy of the defect configuration per vacancy emitted at the

temperature T and the foil surface where the concentration is the equilibrium value c 0 . For a void in equilibrium with its surroundings the free energy F ∼

= 4πr 2 γ s , and since (dF/dn) = (dF/dr)(dr/dn) = (8πrγ s )( /4πr 2 ), where is the atomic volume and n the number

of vacancies in the void, the concentration of vacancies in equilibrium with the void is

c v =c 0 exp [(dF/dn)/kT ] =c 0 exp(2γ s / rkT ).

Assuming spherical diffusion geometry, the diffusion equation may be solved to give the rate of shrinkage of a void as

dr/dt = −(D/r){exp(2 γ s / rkT ) − 1}. (3.22) For large r (>50 nm) the exponential term can be approximated to the first two terms of the series

expansion and equation (3.22) may then be integrated to give r 3 =r 3 i − (6D γ s /kT )t,

(3.23) where r i is the initial void radius at t = 0. By observing the shrinkage of voids as a function of

annealing time at a given temperature (see Figure 3.54), it is possible to obtain either the diffusivity

D or the surface energy γ s . From such observations, γ s for aluminum is shown to be 1.14 J m −2 in the

Crystal defects 151

Figure 3.54 Sequence of micrographs showing the shrinkage of voids in quenched aluminum during isothermal annealing at 170 ◦

C. (a) t = 3 min, ( b) t = 8 min, (c) t = 21 min, (d) t = 46 min, (e) t = 98 min. In all micrographs the scale corresponds to 0.1 µm (after Westmacott, Smallman and Dobson, 1968; courtesy of the Institute of Materials, Minerals and Mining).

C, and D = 0.176 × exp(−1.31 eV/kT). It is difficult to determine γ s for Al by zero creep measurements because of the oxide. This method of obtaining γ s has been applied to other metals and is particularly useful since it gives a value of γ s in the self-diffusion temperature range rather than near the melting point.

temperature range 150–200 ◦

Worked example

Excess vacancies in a crystal may aggregate into dislocation loops or voids. Show that the void is the lower energy configuration at small sizes but above a critical radius has a higher energy than a dislocation loop.

Assume that the surface energy γ s = μb/10, where μ is the shear modulus.

Solution

The energy of the loop E L ≈ 2πR L (μb 2 ), where μb 2 is the dislocation line energy per unit length. This becomes E L

1/2 μ b 3 , since the area of the loop πR 2 = 2π(n/π) 2 L = nb , where n is the number of

vacancies and b 2 is the cross-sectional area of a vacancy.

The energy of a spherical void is E V = 4πR 2 V γ s

= 4π(3n/4π) 2/3 b 2 γ s since the volume of the sphere

4πR 3 V / 3 = nb 3 , where b 3 is the atomic volume.

152 Physical Metallurgy and Advanced Materials Thus, the ratio

E V 4π(3n/4π) 2/3 b 2 γ s

μ b μ b ≈ (n) μ b Typically for solids the surface energy γ s ∼ μb/10.

E = 2π(n/π) 1/2 μ b 3 =2 4 = 1.36(n)

10, i.e. when the radius of the sphere is bigger than about 60 atom spacings.

Thus, E V / E L ≈n 1/6 /

10, and E V > E L when n 1/6 >

3.7.3 Nuclear irradiation effects

3.7.3.1 Behavior of point defects and dislocation loops

Electron microscopy of irradiated metals shows that large numbers of small point defect clusters are formed on a finer scale than in quenched metals, because of the high supersaturation and low

diffusion distance. Bombardment of copper foils with 1.4 ×10 21 38 MeV α-particles m −2 produces about 10 21 m −3 dislocation loops, as shown in Figure 3.55a; a denuded region 0.8 µm wide can also be seen at the grain boundary. These loops, about 40 nm in diameter, indicate that an atomic concentration of ≈1.5 × 10 −4 point defects have precipitated in this form. Heavier doses of α-particle bombardment produce larger diameter loops, which eventually appear as dislocation tangles. Neutron bombardment produces similar effects to α-particle bombardment, but unless the dose is greater than

10 21 neutrons m −2 the loops are difficult to resolve. In copper irradiated at pile temperature the density of loops increases with dose and can be as high as 10 14 m −2 in heavily bombarded metals. The micrographs from irradiated metals reveal, in addition to the dislocation loops, numerous small centers of strain in the form of black dots somewhat less than 5 nm in diameter, which are difficult to resolve (see Figure 3.55a). Because the two kinds of clusters differ in size and distribution, and also in their behavior on annealing, it is reasonable to attribute the presence of one type of defect, i.e. the large loops, to the aggregation of interstitials and the other, i.e. the small dots, to the aggregation of vacancies. This general conclusion has been confirmed by detailed contrast analysis of the defects.

The addition of an extra (1 1 1) plane in a crystal with fcc structure (see Figure 3.56) introduces two faults in the stacking sequence and not one, as is the case when a plane of atoms is removed. In consequence, to eliminate the fault it is necessary for two partial dislocations to slip across the loop, one above the layer and one below, according to a reaction of the form:

The resultant dislocation loop formed is identical to the prismatic loop produced by a vacancy cluster but has a Burgers vector of opposite sign. The size of the loops formed from interstitials increases with the irradiation dose and temperature, which suggests that small interstitial clusters initially form and subsequently grow by a diffusion process. In contrast, the vacancy clusters are much more numerous, and although their size increases slightly with dose, their number is approximately proportional to the dose and equal to the number of primary collisions which occur. This observation supports the suggestion that vacancy clusters are formed by the redistribution of vacancies created in the cascade.

Changing the type of irradiation from electron to light charged particles such as protons, to heavy ions such as self-ions, to neutrons, results in a progressive increase in the mean recoil energy. This results in an increasingly non-uniform point defect generation due to the production of displacement cascades by primary knock-ons. During the creation of cascades, the interstitials are transported outwards (see Figure 3.7), most probably by focused collision sequences, i.e. along a close-packed row

Crystal defects 153

A thin film of copper after bombardment with 1.4 × 10 21 α -particles m −2 . (a) Dislocation loops ( ∼40 nm dia.) and small centers of strain (∼4 nm dia.). ( b) After a 2-hour anneal at 350 ◦

C showing large prismatic loops (after Barnes and Mazey, 1960).

of atoms by a sequence of replacement collisions, to displace the last atom in this same crystallographic direction, leaving a vacancy-rich region at the center of the cascade which can collapse to form vacancy loops. As the irradiation temperature increases, vacancies can also aggregate to form voids.

Frank sessile dislocation loops, double-faulted loops, tetrahedra and voids have all been observed in irradiated metals, but usually under different irradiation conditions. Results from Cu, Ag and Au show that cascades collapse to form Frank loops, some of which dissociate towards stacking fault tetrahedra. The fraction of cascades collapsing to form visible loops, defined as the defect yield, is high, ≈0.5 in Cu to 1.0 in Au irradiated with self-ions. Moreover, the fraction of vacancies taking part in the collapse process, expressed as the cascade efficiency, is also high ( ≈0.3–0.5). Vacancy loops have been observed on irradiation at R.T. in some bcc metals (e.g. Mo, Nb, W, α-Fe). Generally, the loops are perfect with b loops on {1 1 0} but unfault at an early stage because of the high stacking-fault energy. Vacancy loops have also been observed in some cph metals (e.g. Zr and Ti).

Interstitial defects in the form of loops are commonly observed in all metals. In fcc metals Frank loops containing extrinsic faults occur in Cu, Ag, Au, Ni, Al and austenitic steels. Clustering of

154 Physical Metallurgy and Advanced Materials

Normal Extrinsic Intrinsic

Figure 3.56 Single (near top) and double (near middle) dislocation loops (a) and a small double-faulted loop ( b), in proton-irradiated copper. (c) Structure of a double-dislocation loop (after Mazey and Barnes, 1968; courtesy of Taylor & Francis).

interstitials on two neighboring (1 1 1) planes to produce an intrinsically faulted defect may also occur, as shown in Figure 3.56. In bcc metals they are predominantly perfect a/2

The damage produced in cph metals by electron irradiation is very complex and for Zn and Cd (c/a > 1.633) several types of dislocation loops, interstitial in nature, nucleate and grow; thus, c/2 loops, i.e. with b

= [c/2], c loops, (c/2 + p) loops, i.e. with b = 1

and loops consist of [c/2] dislocations, but as they grow a second loop of b = [c/2] forms in the center, resulting in the formation of a [c/2] the nucleation of a dose rates and low temperatures many of the loops facet along

In magnesium, with c/a almost ideal, the nature of the loops is very sensitive to impurities, and

interstitial loops with either b 1 = 3

have been observed in samples with different purity. Double loops with b = (c/2 + p) + (c/2 − p) also form but no c/2 loops have been observed.

In Zr and Ti (c/a < 1.633) irradiated with either electrons or neutrons both vacancy and inter-

stitial loops form on non-basal planes with b 1 =

also form in the temperature range 0.3–0.6T m . The fact that vacancy loops are formed on elec- tron irradiation indicates that cascades are not essential for the formation of vacancy loops. Several factors can give rise to the increased stability of vacancy loops in these metals. One factor is the possibility of stresses arising from oxidation or anisotropic thermal expansion, i.e. interstitial loops are favored perpendicular to a tensile axis and vacancy loops parallel. A second possibility is impurities segregating to dislocations and reducing the interstitial bias.

Crystal defects 155

3.7.3.2 Radiation growth and swelling

In non-cubic materials, partitioning of the loops on to specific habit planes can lead to an anisotropic dimensional change, known as irradiation growth. The aggregation of vacancies into a disk-shaped cavity which collapses to form a dislocation loop will give rise to a contraction of the material in the direction of the Burgers vector. Conversely, the precipitation of a plane of interstitials will result in the growth of the material. Such behavior could account for the growth which takes place in α-uranium single crystals during neutron irradiation, since electron micrographs from thin films of irradiated uranium show the presence of clusters of point defects.

The energy of a fission fragment is extremely high ( ≈200 MeV) so that a high concentration of both vacancies and interstitials might be expected. A dose of 10 24 nm −2 at room temperature causes uranium to grow about 30% in the [0 1 0] direction and contract in the [1 0 0] direction. However,

a similar dose at the temperature of liquid nitrogen produces 10 times this growth, which suggests the preservation of about 10 4 interstitials in clusters for each fission event that occurs. Growth also occurs in textured polycrystalline α-uranium and to avoid the problem a random texture has to be produced during fabrication. Similar effects can be produced in graphite.

During irradiation, vacancies may aggregate to form voids and the interstitials form dislocation loops. The voids can grow by acquiring vacancies which can be provided by the climb of the dislocation loops. However, because these loops are formed from interstitial atoms they grow, not shrink, during the climb process and eventually become a tangled dislocation network.

Interstitial point defects have two properties important in both interstitial loop and void growth. First, the elastic size interaction (see Chapter 6) causes dislocations to attract interstitials more strongly than vacancies and, secondly, the formation energy of an interstitial E i is greater than that of a vacancy

f so that the dominant process at elevated temperatures is vacancy emission. The importance of these factors to loop stability is shown by the spherical diffusion-controlled rate equation:

For the growth of voids during irradiation the spherical diffusion equation

dr

[(2γ s / r) −P] =

]D c exp dt

r + (ρr)

v v − [1 + (Z i

i i −[1 + (ρr)

1/2 ]D c ρ r) 1/2 ]D c 1/2

v 0 kT

has been developed, where c i and c v are the interstitial and vacancy concentrations, respectively, D v and D i their diffusivities, γ s the surface energy and Z i is a bias term defining the preferred attraction of the loops for interstitials.

At low temperatures, voids undergo bias-driven growth in the presence of biased sinks, i.e. dislo- cation loops or network of density ρ. At higher temperatures, when the thermal emission of vacancies becomes important, whether voids grow or shrink depends on the sign of [(2γ s / r) − P]. During neutron irradiation, when gas is being created continuously, the gas pressure P > γ s / r and a flux of gas atoms can arrive at the voids, causing gas-driven growth.

The formation of voids leads to the phenomenon of void swelling and is of practical importance in the dimensional stability of reactor core components. Voids are formed in an intermediate temperature range ≈0.3–0.6T m , above that for long-range single vacancy migration and below that for thermal vacancy emission from voids. To create the excess vacancy concentration it is also necessary to build up a critical dislocation density from loop growth to bias the interstitial flow. The sink strength of the

dislocations, i.e. the effectiveness of annihilating point defects, is given by K 2 i =Z i ρ for interstitials

156 Physical Metallurgy and Advanced Materials and K 2 v =Z v ρ for vacancies, where (Z i −Z v ) is the dislocation bias for interstitials ≈10% and ρ is

the dislocation density. As voids form they also act as sinks, and are considered neutral to vacancies and interstitials, so that K 2 i 2 =K v = 4πr v C v , where r v and C v are the void radius and concentration, respectively. The rate theory of void swelling takes all these factors into account and (1) for moderate dislocation densities as the dislocation structure is evolving, swelling is predicted to increase linearly with irradiation dose, (2) when ρ reaches a quasi-steady state the rate should increase as (dose) 3/2 , and (3) when the void density is very high, i.e. the sink strength of the voids is greater than the sink strength

of the dislocations (K 2 v >> K 2 d ), the rate of swelling should again decrease. Results from electron irradiation of stainless steel show that the swelling rate is linear with dose up to 40 dpa (displacements per atom) and there is no tendency to a (dose) 3/2 law, which is consistent with dislocation structure continuing to evolve over the dose and temperature range examined.

In the fuel element itself, fission gas swelling can occur since uranium produces one atom of gas (Kr and Ze) for every five uranium atoms destroyed. This leads to

≈2 m 3 of gas (STP) per m 3 of U after a ‘burn-up’ of only 0.3% of the uranium atoms.

In practice, it is necessary to keep the swelling small and also to prevent nucleation at grain bound- aries when embrittlement can result. In general, variables which can affect void swelling include alloying elements together with specific impurities, and microstructural features such as precipitates, grain size and dislocation density. In ferritic steels, the interstitial solutes carbon and nitrogen are particularly effective in (1) trapping the radiation-induced vacancies and thereby enhancing recom- bination with interstitials, and (2) interacting strongly with dislocations and therefore reducing the dislocation bias for preferential annihilation of interstitials, and also inhibiting the climb rate of dislo- cations. Substitutional alloying elements with a positive misfit such as Cr, V and Mn with an affinity for C or N can interact with dislocations in combination with interstitials and are considered to have

a greater influence than C and N alone. These mechanisms can operate in fcc alloys with specific solute atoms trapping vacancies and also

elastically interacting with dislocations. Indeed, the inhibition of climb has been advanced to explain the low swelling of Nimonic PE16 nickel-based alloys. In this case precipitates were considered to restrict dislocation climb. Such a mechanism of dislocation pinning is likely to be less effective than solute atoms, since pinning will only occur at intervals along the dislocation line. Precipitates in the matrix which are coherent in nature can also aid swelling resistance by acting as regions of enhanced vacancy–interstitial recombination. TEM observations on θ ′ precipitates in Al–Cu alloys have con- firmed that as these precipitates lose coherency during irradiation, the swelling resistance decreases.

3.7.3.3 Radiation-induced segregation, diffusion and precipitation

Radiation-induced segregation is the segregation under irradiation of different chemical species in an alloy towards or away from defect sinks (free surfaces, grain boundaries, dislocations, etc.). The segregation is caused by the coupling of the different types of atom with the defect fluxes towards the sinks. There are four different possible mechanisms, which fall into two pairs, one pair connected

with size effects and the other with the Kirkendall effect. 12 With size effects, the point defects drag the solute atoms to the sinks because the size of the solute atoms differs from the other types of atom present (solvent atoms). Thus interstitials drag small solute atoms to sinks and vacancies drag large solute atoms to sinks. With Kirkendall effects, the faster diffusing species move in the opposite direction to the vacancy current, but in the same direction as the interstitial current. The former case is usually called the ‘inverse Kirkendall effect’, although it is still the Kirkendall effect, but solute atoms rather than the vacancies are of interest. The most important of these mechanisms, which are

12 The Kirkendall effect is discussed in Chapter 5, Section 5.4.2.

Crystal defects 157

SINK

SINK

INTERSTITIALS VACANCIES

Fast diffusers

Figure 3.57 Schematic representation of radiation-induced segregation produced by interstitial and vacancy flow to defect sinks.

summarized in Figure 3.57, appear to be (1) the interstitial size effect mechanism – the dragging of small solute atoms to sinks by interstitials – and (2) the vacancy Kirkendall effect – the migration away from sinks of fast-diffusing atoms.

Radiation-induced segregation is technologically important in fast breeder reactors, where the high radiation levels and high temperatures cause large effects. Thus, for example, in Type 316 stainless steels, at temperatures in the range 350–650 ◦

C (depending on the position in the reactor), silicon and nickel segregate strongly to sinks. The small silicon atoms are dragged there by interstitials and the slow-diffusing nickel stays there in increasing concentration as the other elements diffuse away by the vacancy inverse Kirkendall effect. Such diffusion (1) denudes the matrix of void-inhibiting silicon and (2) can cause precipitation of brittle phases at grain boundaries, etc.

Diffusion rates may be raised by several orders of magnitude because of the increased concen- tration of point defects under irradiation. Thus, phases expected from phase diagrams may appear at temperatures where kinetics are far too slow under normal circumstances. Many precipitates of this type have been seen in stainless steels that have been in reactors. Two totally new phases have

definitely been produced and identified in alloy systems (e.g. Pd 8 W and Pd 8 V) and others appear likely (e.g. Cu–Ni miscibility gap).

3.7.3.4 Irradiation of ordering alloys

Ordering alloys have a particularly interesting response to the influence of point defects in excess of the equilibrium concentration. Irradiation introduces point defects and their effect on the behavior of ordered alloys depends on two competitive processes, i.e. radiation-induced ordering and radiation- induced disordering, which can occur simultaneously. The interstitials do not contribute significantly to ordering but the radiation-induced vacancies give rise to ordering by migrating through the crystal. Disordering is assumed to take place athermally by displacements. The final state of the alloy at any

158 Physical Metallurgy and Advanced Materials irradiation temperature is independent of the initial condition. At 323 K, Cu 3 Au is fully ordered on

irradiation, whether it is initially ordered or not, but at low temperatures it becomes largely disordered because of the inability of the vacancies to migrate and develop order; the interstitials (E I m ≈ 0.1 eV) can migrate at low temperatures.

Problems

In each of the following four diagrams, mark the Burgers vectors:

u into sheet

u out of sheet

Crystal defects 159

Fcc metals

3.5 Write down the possible Burgers vectors for glide dislocations in an fcc metal.

3.6 For a 1/2[1 ¯1 0] dislocation moving on a (1 1 ¯1) plane, what are the line directions of the screw and edge segments respectively?

3.7 On what planes can a dislocation with Burgers vector 1 2 [1 0 1] glide?

Bcc metals

3.8 What is the line direction of an edge dislocation lying on (1 0 1) whose Burgers vector is 1/2[1 ¯1 ¯1]?

3.9 In what planes can a dislocation with Burgers vector 1 2 [1 1 1] glide?

General

3.10 Complete the following dislocation reactions: (a) a/2[1 1 1] + a/2[1 ¯1 ¯1] → ?

(b) a/6[1 1 ¯1] + a/3[1 1 2] → ? (c) 1/6[¯2 0 2 3] + 1/6[2 0 ¯2 3] → ?

3.11 What are the magnitudes of the following Burgers vectors? (a) a/2[1 1 1] (b) a/6[1 1 2].

3.12 What is the angle between the Burgers vectors a/6[1 1 ¯1] and a/3[1 1 ¯2] (in a cubic crystal), and what is the pole of the plane containing their two directions?

3.13 A sample of nickel is quenched from a temperature of 1500 K to a temperature of 300 K. Estimate the ‘chemical stress’ developed in the nickel by this treatment (GPa). The energy of formation for a vacancy in nickel is 1.4 eV.

3.14 (i) State the Read–Shockley formula for the energy of a tilt boundary. (ii) Show, by calculation, why it is feasible to disregard the term (A − ln θ) when applying

the formula to high-angle boundary surfaces. (Assume that the Burgers vector is 1.3 times larger than the core radius of the constituent dislocations.)

(iii) How does this energy vary with angle of misorientation? Estimate the energy for a large- angle boundary. (iv) Comment on the value of the tilt boundary energy and how it relates to (a) grain boundary energy, (b) surface energy and (c) stacking-fault energy.

3.15 The sketch shows a diagramatric representation of a dissociated edge dislocation, in plan view, on the (1 1 1) plane of a face-centered cubic crystal. Two Shockley partial dislocations (b 2 and

b 3 ) are separated by a stacking fault.

160 Physical Metallurgy and Advanced Materials (a) Given that the separation is four atom spacings calculate the stacking-fault energy of the

material. (Assume the lattice parameter of the material is 0.35 nm and the shear modulus is 40 GN m −2 .)

(b) If the material is assumed to be pure nickel, how would you increase the dissociated width? Justify your conclusion. (c) Write down a possible dissociation reaction in Miller index terms. (d) What would this reaction be in the Thompson tetrahedron notation? (e) It is possible to separate the two partials completely by applying a stress in the plane of the

fault. Indicate whether this stress needs to be parallel or perpendicular to the b 1 direction

and show that this separation will occur if the stress exceeds 5γ/a.

3.16 Given that the anti-phase boundary energy of Ni 3 Al is 175 mJ m −2 , calculate the width of the superdislocation in atom spacings. (Shear modulus = 7.5 × 10 10 Nm −2 ,a = 3.5 × 10 −10 m.)

Further reading

Hirth, J. P. and Lothe, J. (1984). Theory of Dislocations. McGraw-Hill, New York. Hume-Rothery, W., Smallman, R. E. and Haworth, C. W. (1969). Structure of Metals andAlloys, Monograph

No. 1. Institute of Metals. Kelly, A., Groves, G. W. and Kidd, P. (2000). Crystallography and Crystal Defects. Wiley, Chichester. Loretto, M. H. (ed.) (1985). Dislocations and Properties of Real Materials. Institute of Metals, London. Smallman, R. E. and Harris, J. E. (eds) (1976). Vacancies. The Metals Society, London. Sprackling, M. T. (1976). The Plastic Deformation of Simple Ionic Solids. Academic Press, London.

Thompson, N. (1953). Dislocation nodes in fcc lattices. Proc. Phys. Soc., B66, 481.

Chapter 4

Characterization and analysis