Spinodal decomposition

7.2.7 Spinodal decomposition

For any alloy composition where the free energy curve has a negative curvature, i.e. (d 2 G/dc 2 ) < 0, small fluctuations in composition that produce A-rich and B-rich regions will bring about a lowering of the total free energy. At a given temperature the alloy must lie between two points of inflection

(where d 2 G/dc 2 = 0) and the locus of these points at different temperatures is depicted on the phase diagram by the chemical spinodal line (see Figure 7.16). For an alloy c 0 quenched inside this spinodal, composition fluctuations increase very rapidly with time and have a time constant τ

D, where λ is the wavelength of composition modulations in one dimension and D is the interdiffusion coefficient. For such a kinetic process, shown in Figure 7.17, ‘uphill’ diffusion takes place, i.e. regions richer in solute than the average become richer, and poorer

become poorer until the equilibrium compositions c 1 and c 2 of the A-rich and B-rich regions are formed. As for normal precipitation, interfacial energy and strain energy influency the decomposition. During the early stages of decomposition the interface between A-rich and B-rich regions is diffuse

408 Physical Metallurgy and Advanced Materials

Figure 7.17 Composition fluctuations in a spinodal system.

and the interfacial energy becomes a gradient energy, which depends on the composition gradient across the interface according to

G int

depends on the difference in bond energies between like and unlike atom pairs. The coherency strain energy term is related to the misfit ε between regions A and B, where ε = (1/a)da/dc, the fractional change in lattice parameter a per unit composition change, and is given for an elastically isotropic solid by

(7.18) with E Young’s modulus, ν Poisson’s ratio and V the molar volume. The total free energy change

G strain 2 =ε c 2 EV /(1 − v),

arising from a composition fluctuation is therefore

d 2 G 2K

c 2 2 + (2ε EV /(1 − ν)) / 2 (7.19)

dc λ

and a homogeneous solid solution will decompose spinodally provided

(7.20) For λ

−(d 2 G/dc 2 ) > (2K /λ 2 )

+ (2ε 2 EV /(1 − ν)).

= ∞, the condition [(d 2 G/dc 2 )

+ (2ε 2 EV /1 − ν)] = 0 is known as the coherent spinodal, as

shown in Figure 7.16. The λ of the composition modulations has to satisfy the condition

(7.21) and decreases with increasing degree of undercooling below the coherent spinodal line. A λ-value

λ 2 > 2K /[d 2 G/dc 2 + (2ε 2 EV /(1 − ν))]

of 5–10 nm is favored, since shorter λ-values have too sharp a concentration gradient and longer values have too large a diffusion distance. For large misfit values, a large undercooling is required to

Mechanical properties II – Strengthening and toughening 409 overcome the strain energy effect. In cubic crystals, E is usually smaller along

the high strain energy is accommodated more easily in the elastically soft directions, with composition modulations localized along this direction.

Spinodal decompositions have now been studied in a number of systems such as Cu–Ni–Fe, Cu–Ni–Si, Ni–12Ti and Cu–5Ti exhibiting ‘side-bands’ in X-ray small-angle scattering, satellite spots in electron diffraction patterns and characteristic modulation of structure along tron micrographs. Many of the alloys produced by splat cooling might be expected to exhibit spinodal decomposition, and it has been suggested that in some alloy systems GP zones form in this way at high supersaturations, because the GP zone solvus (see Figure 7.1) gives rise to a metastable coherent miscibility gap.

The spinodally decomposed microstructure is believed to have unusually good mechanical stability under fatigue conditions.