Particle coarsening

7.2.6 Particle coarsening

With continued ageing at a given temperature, there is a tendency for the small particles to dissolve and the resultant solute to precipitate on larger particles, causing them to grow, thereby lowering the total interfacial energy. This process is termed particle coarsening, or sometimes Ostwald ripening.

The driving force for particle growth is the difference between the concentration of solute (S r ) in equilibrium with small particles of radius r and that in equilibrium with larger particles. The variation of solubility with surface curvature is given by the Gibbs–Thomson or Thomson–Freundlich equation:

ln (S r / S) = 2γ /kTr, (7.11) where S is the equilibrium concentration, γ the particle/matrix interfacial energy and the atomic

volume; since 2γ << kTr, then S r = S[1 + 2γ /kTr]. To estimate the coarsening rate of a particle it is necessary to consider the rate-controlling process for material transfer. Generally, the rate-limiting factor is considered to be diffusion through the matrix, and the rate of change of particle radius is then derived from the equation:

4πr 2 (dr/dt)

= D4πR 2 (dS/dR),

where dS/dR is the concentration gradient across an annulus at a distance R from the particle center. Rewriting the equation after integration gives

(7.12) where S a is the average solute concentration a large distance from the particle and D is the solute

dr/dt = −D(S r −S a )/r,

diffusion coefficient. When the particle solubility is small, the total number of atoms contained in particles may be assumed constant, independent of particle size distribution. Further consideration shows that

(S a −S r ) = {2γ S/kT }[(1/r) − (1/r)] and combining with equation (7.11) gives the variation of particle growth rate with radius according to dr/dt = {2DSγ /kTr}[(1/r) − (1/r)].

(7.13) This function is plotted in Figure 7.14, from which it is evident that particles of radius less than r

are dissolving at increasing rates with decreasing values of r. All particles of radius greater than r are growing but the graph shows a maximum for particles twice the mean radius. Over a period of time the number of particles decreases discontinuously when particles dissolve, and ultimately the system would tend to form one large particle. However, before this state is reached the mean radius r increases and the growth rate of the whole system slows down.

A more detailed theory than that, due to Greenwood, outlined above has been derived by Lifshitz and Slyozov, and by Wagner, taking into consideration the initial particle size distribution. They show that the mean particle radius varies with time according to

r 3 −r 3 0 = Kt, (7.14) where r 0 is the mean particle radius at the onset of coarsening and K is a constant given by K = 8DSγ /9kT .

Mechanical properties II – Strengthening and toughening 405

dt dr/

Figure 7.14 The variation of growth rate dr/dt with particle radius r for diffusion-controlled growth, for two values of γ. The value of γ for the lower curve is 1.5 times that for the upper curve. Particles of radius equal to the mean radius of all particles in the system at any instant are neither growing nor dissolving. Particles of twice this radius are growing at the fastest rate. The smallest

particles are dissolving at a rate approximately proportional to r 2 (after Greenwood, 1968; courtesy of Institute of Materials, Minerals and Mining).

This result is almost identical to that obtained by integrating equation (7.13) in the elementary theory and assuming that the mean radius is increasing at half the rate of that of the fastest growing particle.

Coarsening rate equations have also been derived assuming that the most difficult step in the process is for the atom to enter into solution across the precipitate/matrix interface; the growth is then termed interface controlled. The appropriate rate equation is

dr/dt = −C(S r −S a ) and leads to a coarsening equation of the form r 2 −r 2 0 = (64CSγ t/8kT ),

(7.15) where C is some interface constant.

Measurements of coarsening rates carried out so far support the analysis basis on diffusion control of the particle growth. The most detailed results have been obtained for nickel-based systems, particularly the coarsening of γ ′ (Ni 3 Al–Ti or Si), which show a good r 3 versus t relationship over a wide range of temperatures. Strains due to coherency and the fact that γ ′ precipitates are cube shaped do not seriously affect the analysis in these systems. Concurrent measurements of r and the solute concentration in the matrix during coarsening have enabled values for the interfacial energy ≈13 mJ m −2 to be determined. In other systems the agreement between theory and experiment is generally less precise, although generally the cube of the mean particle radius varies linearly with time, as shown in Figure 7.15 for the growth of Mn precipitates in an Mg–Mn alloy.

406 Physical Metallurgy and Advanced Materials

G 350⬚C

Annealing time (h)

Figure 7.15 The variation of r 3 with time of annealing for manganese precipitates in a magnesium matrix (after Smith, 1967).

Because of the ease of nucleation, particles may tend to concentrate on grain boundaries, and hence grain boundaries may play an important part in particle growth. For such a case, the Thomson– Freundlich equation becomes

ln (S r / S) = (2γ − γ g ) /kTx,

where γ g is the grain boundary energy per unit area and 2 × the particle thickness, and their growth follows a law of the form:

r 4 f 4 −r 0 = Kt, (7.16) where the constant K includes the solute diffusion coefficient in the grain boundary and the boundary

width. The activation energy for diffusion is lower in the grain boundary than in the matrix and this leads to a less strong dependence on temperature for the growth of grain boundary precipitates. For this reason their preferential growth is likely to be predominant only at relatively low temperature.

Worked example

During the ageing of a Cu–Co alloy, the spherical Co precipitates coarsen such that they double their initial size after 14 hours at 527 ◦

C. Calculate the activation energy for the process, which may be assumed to be volume diffusion controlled (R = 8.31 J mol −1 K −1 ).

C and treble their initial size after 8 hours at 577 ◦

Solution

The Ostwald ripening equation for coarsening is r 3 t 3 −r o = 8γcv m Dt/9RT , where r t and r o are the mean particle radius at time t and 0 respectively, γ is the surface free energy of the particle/matrix interface, v m is the molar volume of precipitate, c the equilibrium solute concentration of the matrix and D the diffusivity of the diffusing species. Over the temperature range of the experiment γ, c and v m can be assumed independent of temperature.

Thus, at T 1 = 800 K, r t = 2r o ,t 1 = 5 × 10 4 s

T 2 = 850 K, r t = 3r o ,t

2 = 2.88 × 10 s.

Hence, r 3 o (8

3 − 1) = 7r 4

o = constant × D 1 × 5 × 10 / 800 for T 1 = 800 K

(i)

Mechanical properties II – Strengthening and toughening 407

) 1 G (T

d 2 G dc 2

Free energy

c 1 c 2 Chemical

spinodal Composition c

Figure 7.16 Variation of chemical and coherent spinodal with composition.

and r 3 o (27

3 − 1) = 26r 4 o = constant × D 2 × 2.88 × 10 / 850 for T 2 = 850 K. (ii) From (i), D 1 = 56r 3 o / constant × 500.

From (ii), D 2 = 221r 3 o / constant × 288. Thus, D 1 / D 2 = 0.146, and from Arrenhius rate theory

D 1 / D 2 = exp [−(Q/R)(1/T 1 − 1/T 2 )] ln (D 1 / D 2 ) = −(Q/R)(1/T 1 − 1/T 2 )

ln (0.146) = −(Q/8.31)(1/800 − 1/850) Q

= 2.2 × 10 5 J mol −1 .