The solute diffusion coefficient required for clustering is anomalously high in quenched dilute alloys* There have been two basic attempts to explain this effectj in one it is
proposed that diffusion takes place along dislocation lines and in the other the effect is attributed to a non-equilibrium concentration of vacancies. The dislocation mechanism gives a reasonable explanation of the absolute value of the anomalous diffusion coefficient, but does not-explain why the coefficient should be so sensitive to quenching rate, homogenization
temperature,etc. Consequently this theory has been superceded by the excess vacancy mechanism.
The excess vacancy theory explains most of the experimental facts concerning the kinetics of clustering. It predicts that diffusion is governed by a vacancy concentration which would
be in equilibrium at the solution temperature, Tg, rather than at, T^, the ageing temperature. For the diffusion of Cu in Al- it has been shov/n that:-
—E • — E
DCu = A.exp ( jf^J.exp ( f # - ) (l)
A XI
where Ep and E^ are the activation energies for the formation and migration of vacancies respectively, and A is a constant. Equation (l) explains why the rate of quenching has such a marked effect on the kinetics of clustering. If the quench is slow some vacancies have time to migrate to sinks during quenching and the retained concentration is not associated with but with some lower temperature. Also this equation predicts that the clustering process should be governed by an activation energy, E^, for the migration of vacancies.
These clusters are no different from zones except that during or immediately after quenching, the solute atoms
segregate to form clusters, but these clusters grow until they give rise to the x-ray diffraction effects typical of G.P.
(32)
zones. When there is no atomic size difference between the solute and the solvent, the shape of the zones is
spherical but where there is a size difference the zones, and probably the first clusters, are assymetrical and may take the form of plates and rods^ .
In the case of alloys showing precipitation in the matrix, the clusters or zones may not form above a well defined
temperature, called the G.P. solvus temperature(^3). Below this temperature G.P. zones form spontaneously and above
this temperature zones are -unstable and are replaced by hetro- geneous precipitation. This effect has been attributed to the energy barrier for nucleation of zones, which is zero beloY/ the G.P. solvus temperature, and the reaction rate is then solely controlled by solute diffusion. Above this temperature, the retardation in the rate of zone formation indicates that the energy barrier becomes positive and increases progressively with increasing temperatures.
2.2.2.2 Nucleation
precipitation because the energy required for solute atoms to add to the clusters is much less than for their precipita tion at new nucleation sites. These nuclei contribute stable regions having a chemical composition different from the
composition of the matrix. As precipitation proceeds the aggregates grow and may undergo internal ordering to produce a precipitate with a crystal structure quite different from that of the matrix.
The shape of the precipitate particles and the nature of the interface between the precipitate and the matrix are of interest and have a great influence on the mechanical properties of the final structure. The shape of the
precipitate is greatly influenced by the nature of the inter-
(
52)face between the precipitate and the matrix.'* 7 A fully coherent interface is defined when the crystals are placed in contact such that the plane of the atoms constituting the interface has an atomic arrangement, irrespective of the
chemical species of the atoms, which is common to both crystal structures. In general , the interplanar spacing of the atoms in a coherent interface is not quite the same in both the
matrix and the precipitate, so that only when the interface is*small in area can it be fully coherent, and misfit must then be accommodated by elastic strains. If the interface becomes large in area the strain energy can be reduced by the introduction of dislocations lying in the interface.
Such an interface is not fully coherent . Thus a completely coherent precipitate is one in which all interfaces with the matrix are coherent and the bravais lattices in the two crystal
structures are identical, if differences in chemical species, and consequent changes in atomic spacings are neglected, e.g. gamma prime in austenite matrices. A partially coherent
precipitate is defined as one which has an interface in which there are interface dislocations. A non-coherent precipitate is one in which none of the interfaces Y/ith matrix exhibit even partial coherency.
Nabarro showed that if the precipitate remains completely coherent with the matrix, the strain energy of the particle
( ^5)
sphere of unstrained radius (l+S)t*0 is inserted into a
spherical hole of radius f* in the matrix, the total elastic energy of a sphere of volume v = h/3 K and bulk modulus K, in the matrix of rigidity modulus G , is:-
w = (2)
The energy thus increases as the volume of the precipitate particle increases® If the precipitate breaks away from the matrix and forms a non-coherent interface of such a type that
the shear strain in the precipitate vanishes, and if all the strain occurs in the matrix, the energy is given by:-
W = 6 6 v & 2 £ (c/a) (3)
where f(c/a) is a function of the shape of the spheroid® A more recent calculation by Brown and Ham (^6) on effect of coherent precipitates on the shear stress, *C gives:-
■c = 4.ig € 3/2 f1/2 r^-)1/2 ... (u)
, 1 for small precipitates
JL v.3 V 4
and r = 0.7Gif2 ( ^ T 3 ) ... (5)
' ° * for largejprecipitates. where f = the volume fraction of precipitate,
b = Burger*s Vector of dislocations
and £ = is a measure of the Shear Strain due to mis-match of the precipitate with the matrix®
2®2®2®3 Growth of the Precipitates.
As the precipitate grov/s, so does the localised internal strain® In time, these strains build up to the . point where local shearing takes place and the precipitate becomes an isolated particle of the second phase. When shearing occurs, there is a definite release of internal strain and a decrease
in the size of the strain affected region Subsequent
agglomeration of the second phase particles further reduces the effectiveness of the strained regions because fewer precipitates are present, and dislocations can travel greater distances
( X~?)
It has been shownw,/ that the isothermal growth of garnma prime precipitates in nickel-base alloys can be represented by a relationship of the form
D » Q t* + C (6)
where D = diameter of the precipitate in angstrom units, t is the time and 61 and n are constants. Mitchell/ 7Q \ has stated that the value of n is about 1/3. The growth
of the precipitates is diffusion controlled, and the rate of growth increases with increasing supersaturation and
T 39)