Solute–dislocation interaction

6.4.7 Solute–dislocation interaction

Iron containing carbon or nitrogen shows very marked yield point effects and there is a strong elastic interaction between these solute atoms and the dislocations. The solute atoms occupy interstitial

316 Physical Metallurgy and Advanced Materials

Figure 6.27 Weak-beam micrographs showing separation of superdislocation partials in

Cu 3 Au. (a) As deformed. (b) After ageing at 225 ◦

C (after Morris and Smallman, 1975).

sites in the lattice and produce large tetragonal distortions, as well as large-volume expansions. Consequently, they can interact with both shear and hydrostatic stresses and can lock screw as well as edge dislocations. Strong yielding behavior is also expected in other bcc metals, provided they contain interstitial solute elements. On the other hand, in the case of fcc metals the arrangement of lattice positions around either interstitial or substitutional sites is too symmetrical to allow a solute atom to produce an asymmetrical distortion, and the atmosphere locking of screw dislocations, which requires a shear stress interaction, would appear to be impossible. Then by this argument, since the screw dislocations are not locked, a drop in stress at the yield point should not be observed. Nevertheless, yield points are observed in fcc materials and one reason for this is that unit dislocations in fcc metals dissociate into pairs of partial dislocations which are elastically coupled by a stacking fault. Moreover, since their Burgers vectors intersect at 120 ◦ there is no orientation of the line of the pair for which both can be pure screws. At least one of them must have a substantial edge component, and a locking of this edge component by hydrostatic interactions should cause a locking of the pair, although it will undoubtedly be weaker.

In its quantitative form the theory of solute atom locking has been applied to the formation of an atmosphere around an edge dislocation due to hydrostatic interaction. Since hydrostatic stresses are scalar quantities, no knowledge is required in this case of the orientation of the dis- location with respect to the interacting solute atom, but it is necessary in calculating shear stresses

interactions. 4 Cottrell and Bilby have shown that if the introduction of a solute atom causes a volume

4 To a first approximation a solute atom does not interact with a screw dislocation, since there is no dilatation around the screw; a second-order dilatation exists, however, which gives rise to a non-zero interaction falling off with distance

from the dislocation according to 1/r 2 . In real crystals, anisotropic elasticity will lead to first-order size effects, even with screw dislocations, and hence a substantial interaction is to be expected.

Mechanical properties I 317

interaction energy is

V (6.9)

point (R, θ) from a positive edge dislocation is b(1 − 2ν) × sin θ/2πR(1 − ν), and substituting K = 2μ(1 + ν)/3(1 − 2ν), where μ is the shear modulus and ν Poisson’s ratio, we get the expression:

V (R,θ) (6.10) = A sin θ/R.

This is the interaction energy at a point whose polar coordinates with respect to the center of the dislocation are R and θ. We note that V is positive on the upper side (0 < θ < π) of the

itative picture of a large atom being repelled from the compressed region and attracted into the expanded one.

It is expected that the site for the strongest binding energy V max will be at a point θ = 3π/2, R =r 0 −29 Nm 2 and V max ≈ 1 eV for carbon or nitrogen in α-iron. This value is almost certainly too high because of the limitations of the interaction energy equation in describing conditions near the center of a dislocation, and a more realistic value obtained from experiment (e.g. internal friction experiments)

is V max 1 ≈ 2 to 3 4 also easier to calculate from lattice parameter measurements. Thus, if r and r(1 + ε) are the atomic

4πr 3 ε and equation (6.10) becomes

V = 4(1 + ν)μbεr 3 sin θ/3(1 − ν)R (6.11) = A sin θ/R.

Taking the known values μ = 40 GN m −2 ,ν = 0.36, b = 2.55 × 10 −10 m, r 0 and ε = 0.06, we find

A ≈ 5 × 10 −30 Nm 2 , which gives a much lower binding energy, V max 1 = 8 eV. The yield phenomenon is particularly strong in iron because an additional effect is important; this concerns the type of atmosphere a dislocation gathers round itself, which can be either condensed or dilute. During the strain-ageing process, migration of the solute atoms to the dislocation occurs and two important cases arise. First, if all the sites at the center of the dislocation become occupied the atmosphere is then said to be condensed; each atom plane threaded by the dislocation contains one solute atom at the position of maximum binding, together with a diffuse cloud of other solute atoms further out. If, on the other hand, equilibrium is established before all the sites at the center are saturated, a steady state must be reached in which the probability of solute atoms leaving the center can equal the probability of their entering it. The steady-state distribution of solute atoms around the dislocations is then given by the relation

C (R,θ) =c 0 exp [V (R,θ) /kT ],

where c 0 is the concentration far from a dislocation, k is Boltzmann’s constant, T is the absolute temperature and C the local impurity concentration at a point near the dislocation where the binding energy is V . This is known as the dilute or Maxwellian atmosphere. Clearly, the form of an atmosphere

318 Physical Metallurgy and Advanced Materials will be governed by the concentration of solute atoms at the sites of maximum binding energy V max ,

and for a given alloy (i.e. c 0 and V max fixed) this concentration will be

C V max =c 0 exp(V max /kT )

(6.12) as long as C V max is less than unity. The value of C V max depends only on the temperature, and as the

temperature is lowered C V max will eventually rise to unity. By definition the atmosphere will then have passed from a dilute to a condensed state. The temperature at which this occurs is known as the condensation temperature T c , and can be obtained by substituting the value C V max = 1 in equation (6.12) when

T c =V max /k ln (1/c 0 ).

(6.13) Substituting the value of V max for iron, i.e. 1 2 eV, in this equation we find that only a very small

concentration of carbon or nitrogen is necessary to give a condensed atmosphere at room temperature, and with the usual concentration strong yielding behavior is expected up to temperatures of about 400 ◦ C.

In the fcc structure, although the locking between a solute atom and a dislocation is likely to

be weaker, condensed atmospheres are still possible if this weakness can be compensated for by sufficiently increasing the concentration of the solution. This may be why examples of yielding in fcc materials have been mainly obtained from alloys. Solid solution alloys of aluminum usually contain less than 0.1 at.% of solute element, and these show yielding in single crystals only at low temperature (e.g. liquid nitrogen temperature, −196 ◦ C), whereas supersaturated alloys show evidence of strong yielding even in polycrystals at room temperature; copper dissolved in aluminum has a misfit value ε

≈ 0.12, which corresponds to V 1 max =

4 eV, and from equation (6.13) it can be shown that a 0.1 at.% alloy has a condensation temperature T c = 250 K. Copper-based alloys, on the other hand, usually

form extensive solid solutions and, consequently, concentrated alloys may exhibit strong yielding phenomena.

The best-known example is α-brass and, because V max 1 ≈ 8 eV, a dilute alloy containing 1 at.% zinc has a condensation temperature T c ≈ 300 K. At low zinc concentrations (1–10%) the yield point in brass is probably solely due to the segregation of zinc atoms to dislocations. At higher concentrations, however, it may also be due to short-range order.