Grain growth

6.8.4 Grain growth

When primary recrystallization is complete (i.e. when the growing crystals have consumed all the strained material), the material can lower its energy further by reducing its total area of grain surface.

With extensive annealing it is often found that grain boundaries straighten, small grains shrink and larger ones grow. The general phenomenon is known as grain growth, and the most important factor governing the process is the surface tension of the grain boundaries. A grain boundary has a surface tension, T ( = surface free energy per unit area), because its atoms have a higher free energy than those within the grains. Consequently, to reduce this energy a polycrystal will tend to minimize the area of its grain boundaries and, when this occurs, the configuration taken up by any set of grain boundaries (see Figure 6.53) will be governed by the condition that

(6.45) Most grain boundaries are of the large-angle type with their energies approximately independent

T A / sin A =T B / sin B =T C / sin C.

of orientation, so that for a random aggregate of grains T A =T B =T C and the equilibrium grain boundary angles are each equal to 120 ◦ . Figure 6.53b shows an idealized grain in two dimensions surrounded by others of uniform size, and it can be seen that the equilibrium grain shape takes the form of a polygon of six sides with 120 ◦ inclusive angles. All polygons with either more or less than this number of sides cannot be in equilibrium. At high temperatures where the atoms are mobile,

356 Physical Metallurgy and Advanced Materials

Figure 6.53 (a) Relation between angles and surface tensions at a grain boundary triple point. (b) Idealized polygonal grain structure.

ur

Figure 6.54 Diagram showing the drag exerted on a boundary by a particle.

a grain with fewer sides will tend to become smaller, under the action of the grain boundary surface tension forces, while one with more sides will tend to grow. Second-phase particles have a major inhibiting effect on boundary migration and are particularly effective in the control of grain size. The pinning process arises from surface tension forces exerted by the particle–matrix interface on the grain boundary as it migrates past the particle. Figure 6.54 shows that the drag exerted by the particle on the boundary, resolved in the forward direction, is

F = πrγ sin 2θ, where γ is the specific interfacial energy of the boundary; F =F max = πrγ when θ = 45 ◦ . Now if

there are N particles per unit volume, the volume fraction is 4πr 3 N /3 and the number n intersecting unit area of boundary is given by

n = 3f /2πr 2 . (6.46) For a grain boundary migrating under the influence of its own surface tension the driving force is

2γ/R, where R is the minimum radius of curvature and, as the grains grow, R increases and the driving force decreases until it is balanced by the particle drag, when growth stops. If R ∼ d, the mean grain diameter, then the critical grain diameter is given by the condition

nF ≈ 2γ/d crit or

d crit 2 ≈ 2γ(2πr / 3f πrγ) = 4r/3f . (6.47)

Mechanical properties I 357 This Zener drag equation overestimates the driving force for grain growth by considering an isolated

spherical grain. A heterogeneity in grain size is necessary for grain growth, and taking this into account gives a revised equation:

d crit ≈ − , (6.48) 3f 2 Z

where Z is the ratio of the diameters of growing grains to the surrounding grains. This treat- ment explains the successful use of small particles in refining the grain size of commercial alloys.

During the above process growth is continuous and a uniform coarsening of the polycrystalline aggregate usually occurs. Nevertheless, even after growth has finished the grain size in a specimen which was previously severely cold-worked remains relatively small, because of the large number of nuclei produced by the working treatment. Exaggerated grain growth can often be induced, however, in one of two ways, namely: (1) by subjecting the specimen to a critical strain-anneal treatment or (2) by a process of secondary recrystallization. By applying a critical deformation (usually a few percent strain) to the specimen the number of nuclei will be kept to a minimum, and if this strain is followed by

a high-temperature anneal in a thermal gradient some of these nuclei will be made more favorable for rapid growth than others. With this technique, if the conditions are carefully controlled, the whole of the specimen may be turned into one crystal, i.e. a single crystal. The term secondary recrystallization describes the process whereby a specimen which has been given a primary recrystallization treatment at a low temperature is taken to a higher temperature to enable the abnormally rapid growth of a few grains to occur. The only driving force for secondary recrystallization is the reduction of grain boundary free energy, as in normal grain growth, and, consequently, certain special conditions are necessary for its occurrence. One condition for this ‘abnormal’ growth is that normal continuous growth is impeded by the presence of inclusions, as is indicated by the exaggerated grain growth of tungsten wire containing thoria, or the sudden coarsening of deoxidized steel at about 1000 ◦ C.

A possible explanation for the phenomenon is that in some regions the grain boundaries become free (e.g. if the inclusions slowly dissolve or the boundary tears away) and as a result the grain size in such regions becomes appreciably larger than the average (Figure 6.55a). It then follows that the grain boundary junction angles between the large grain and the small ones that surround it will not satisfy the condition of equilibrium discussed above. As a consequence, further grain boundary movement to achieve 120 ◦ angles will occur, and the accompanying movement of a triple junction point will

be as shown in Figure 6.55b. However, when the dihedral angles at each junction are approximately 120 ◦ a severe curvature in the grain boundary segments between the junctions will arise, and this leads to an increase in grain boundary area. Movement of these curved boundary segments towards their centers of curvature must then take place and this will give rise to the configuration shown in Figure 6.55c. Clearly, this sequence of events can be repeated and continued growth of the large grains will result.

The behavior of the dispersed phase is extremely important in secondary recrystallization and there are many examples in metallurgical practice where the control of secondary recrystallization with dispersed particles has been used to advantage. One example is in the use of Fe–3% Si in the production of strip for transformer laminations. This material is required with (1 1 0) [0 0 1] ‘Goss’ texture because of the [0 0 1] easy direction of magnetization, and it is found that the presence of MnS particles favors the growth of secondary grains with the appropriate Goss texture. Another example is in the removal of the pores during the sintering of metal and ceramic powders, such as alumina and metallic carbides. The sintering process is essentially one of vacancy creep involv- ing the diffusion of vacancies from the pore of radius r to a neighboring grain boundary, under a

358 Physical Metallurgy and Advanced Materials

(c) Figure 6.55 Grain growth during secondary recrystallization.

(a)

(b)

Critical strain-anneal region

Secondary recrystallization region

Grain size

0 3 6 10 20 40 60 80 % Deformation

Figure 6.56 Relation between grain size, deformation and temperature for aluminum (after Burgers, courtesy of Akademie-Verlags-Gesellschaft).

driving force 2γ s / r, where γ s is the surface energy. In practice, sintering occurs fairly rapidly up to about 95% full density because there is a plentiful association of boundaries and pores. When the pores become very small, however, they are no longer able to anchor the grain boundaries against the grain growth forces, and hence the pores sinter very slowly, since they are stranded within the grains some distance from any boundary. To promote total sintering, an effective dispersion is added. The dispersion is critical, however, since it must produce sufficient drag to slow down grain growth, during which a particular pore is crossed by several migrating boundaries, but not sufficiently large to give rise to secondary recrystallization when a given pore would be stranded far from any boundary.

The relation between grain size, temperature and strain is shown in Figure 6.56 for commercially pure aluminum. From this diagram it is clear that either a critical strain-anneal treatment or a secondary recrystallization process may be used for the preparation of perfect strain-free single crystals.