# Aspects of kinetic theory

## Grain growth

A real, three-dimensional grain structure cannot ever be in equilibrium because the space-filling grain-shape prevents the balancing of interfacial tensions. A uniform, hexagonal grain structure in two dimensions can, on the other hand, be metastable because the boundary triple-points exhibit three-fold symmetry so the tensions are balanced, assuming that the interfacial energy is identical for all boundaries. In a single-phase material, there will exist a distribution of grains sizes with some grains having a greater number of sides than others; the distribution of sizes is in general unimodal with a maximum that is twice as large as the mean value [1]. In such a structure, annealing enables the larger grains to grow at the expense of those that are smaller, but the unimodal distribution of sizes is maintained even though the mean size increases. The process is driven entirely by the excess energy present in the structure due to interfaces. On a local scale, grain boundaries tend to migrate towards their centre of curvature as atoms located at a curved boundary move into positions where they have more correctly positioned near-neighbours. The attempt at balancing tensions at grain boundary junctions leads to curvature that in turn promotes the growth of the larger grains. Figure 13.1.

Figure 13.1 (a) An exaggerated illustration in which the grains all have flat faces so interfacial tensions are not balanced at grain boundary junctions. The dashed circle highlights one such junction where if the boundary energies are all identical, then the forces at the junction are not balanced, (b) A relaxed grain configuration where boundary segments are curved to help maintain a semblance of balance at grain boundary junctions. They therefore migrate towards their centres of curvature.

When the grain size L is measured as a mean lineal intercept, the boundary surface per unit volume is given by 2/L, so the excess energy locked up in the form of grain boundaries is 2a/L. This excess energy drives the coarsening of the three-dimensional grain structure; cr is the grain boundary energy per unit area, assumed to be single-valued.

The microstructure might contain obstacles to grain boundary motion, for example, precipitates. Suppose that there is a random array of particles, volume fraction Vv with (Vv uniformly-sized, spherical particles per unit volume, each of radius /- then N = 3VV/Am3 and the number of particles intersected by a unit area of boundary is 2WVv so the force opposing boundary motion is 3oVy/2r. This force is the Zener drag on the boundary [reviewed in 2]. A limiting grain size Llim can be defined when the particle pinning force equals the driving force for grain growth, i.e., when 2a/L = 3Vy/2r, so that Тцт = 4r/3Vv.

For small driving forces, the average boundary migration velocity is equal to a boundary mobility Мь multiplied by the net driving force (Equation 4.64), an approximation justified at small driving forces;

If the boundary velocity is written as dL/dt and the temperature dependence of the mobility can be separated as Mb = Mboep(-Q/RT), then it follows that [3]

where t is the time during isothermal growth and Q is an activation energy for grain growth. If Llim = oo then integration would lead to a relationship where I varies with ft assuming that the initial grain size before coarsening is neglected.1 But in general, integration of Equation 13.1 gives [3]

where La is the initial grain size. When considering austenite grain growth in large components, the time taken to reach the heat-treatment temperature can be taken into account. The steel becomes fully austenitic at a temperature Асз so La is set to that austenite grain size that exists at that temperature. During continuous heating followed by isothermal holding at a temperature T for time t, the term on the right hand side of Equation 13.2 becomes [4]

where T is the heating rate between Act, and V. The activation energy Q is about 200 kJ mol-1, somewhat less than that for the self-diffusion of iron; much larger values have been reported but are a consequence of overfitting to limited experimental data [4].

There are circumstances in which some grains grow much more rapidly than the general coarsening described above. Figure 13.2 illustrates a steel containing boundary-pinning AIN particles that are stable at the lower austenitisation temperature of 840 °C, but begin to dissolve in the austenite at 940 °C. In this latter regime of borderline particle stability, some regions will become depleted of particles before others, where relatively rapid grain growth can occur locally, leading to a bi- modal distribution of grain sizes. Other heterogeneities that promote abnormal grain growth include anisotropy in crystal orientations, in boundary energy or boundary mobility. The conditions lead-

Figure 13.2 Austenite grain growth in a nuclear pressure vessel steel containing a small fraction of aluminium nitride precipitates. The austenite grains are revealed using thermal grooving that occurs during austenitisation as grain boundary and surface tensions are balanced, (a) Following austenitisation at 840 °C, (b) following austenitisation at 940 °C. (c, d) Respective distributions of lineal intercepts [4], showing that there is a bimodal distribution for the sample austenitised at 940 °C.

ing to abnormal grain growth have been modelled [1,5], suggesting that grains with a size about 1.4 times the average will tend to grow abnormally. The practical application of this requires some mechanism by which such a size anomaly can appear in the microstructure.

## Recrystallisation

In normal circumstances, any reduction in the density of dislocations, introduced during plastic deformation, is small during the process of recovery. The deformed grain structure is also largely unaffected by recovery. It takes the growth of new' grains to initiate a much larger change, i.e., recrystallisation. New grains are stimulated in regions w'here the dislocation density is large (Figure 13.3). This can happen when an existing grain boundary bow's into the grain containing a somewhat greater density of dislocations, in w'hich case the crystallographic orientation of the grain with the lower dislocation density is maintained in the recrystallised region, with consequences on the development of a recrystallisation texture.

If the steel contains relatively hard particles that resist changes in shape, then deformation can lead to plastic strain gradients around the particles. When these gradients are large, they can lead to particle-stimulated recrystallisation where the new grain may have a large misorientation relative to its surroundings. Large particles are more effective since there are greater deformation gradients around them and hence are expected to be more effective in inducing recrystallisation. The deformation gradients extend approximately to the size of the coarse particles [6].

Figure 13.3 Nucleation of recrystallisation, (a) by grain boundary bowing that propagates the orientation of the grain on the left, (b) particle stimulated nucleation. which may lead to a highly misoriented grain, (c) Development of misorientation by polygonisation. (d) Variation in recrystallised grain size as a function of the amount of deformation prior to isothermal annealing.

Both of the mechanisms described require a certain misorientation to develop within the deformed matrix in order to define the genesis of a recrystallisation nucleus. Suppose that nucleation begins in a jumble of dislocations. The local rearrangement of free dislocations can lead to the creation of a region that is essentially free from dislocations. Figure 13.3c. If it is assumed that a new grain forms when a region accumulates a misorientation в = 10° ~ 0.2 rad with its neighbours and a size Zi, both arrived at by polygonisation, then:

so that zi = 15 x 10 l0m. Given that z is typically 0.1-1 pm, the critical dislocation density required to generate the misorientation, = 1 jz2 Ю15 1016 m~2. The actual dislocation den?sity required has to be somewhat larger if some of the defects are annihilated during recovery. Large dislocation density differences are usually only to be found in localised regions.

A greater nucleation rate leads ultimately to a finer recrystallised grain size (Figure 13.3d) during isothermal annealing. There is a level of deformation below which recrystallisation does not occur because recovery processes reduce the defect density. While the extent of the plastic strain prior to annealing is a key parameter influencing recrystallisation behaviour, other factors affect the nature of the defects introduced during deformation:

• changes to the shapes of the grain following deformation. For example, during plane-strain deformation (rolling), austenite grains are deformed in the rolling direction more severely than in other directions. This pancaking that increases the amount of boundary per unit volume from SV0 in the undeformed state to Sy following plastic deformation, which adds to the stored energy of the material. For grains that initially are equiaxed, in the form of space-filling tetrakaidecahedra [7, 8|:

where t)n and r33 are the principal distortions with the true strains along the principal directions given by e„ = In t]„; L is the grain edge length per unit volume, with initial value Ly0.

Because plastic deformation is not homogeneous, boundaries are roughened by slip [9) and heterogeneous slip introduces deformation bands. There may also exist annealing twins within the austenite grains. These effects are not predictable, but certainly affect the stored energy. When considering austenite grains, the effects are represented empirically; the additional Sy due to twins and bands is taken to be proportional to e~ [ 10]. If the flow stress dp is known, then the dislocation density is often taken to be proportional to (ор/Еф)2, with the proportionality constant containing the Taylor factor and an empirical constant [11].

• If a hole in a matrix is sheared, then the distortion accompanying the hole can be restored to its original shape by an appropriate deformation of the matrix, or by local matrix rotations [12, 13]; this process in effect models what happens when a material containing a rigid inclusion is deformed. The lattice rotations may be expressed in the form of subgrains which deviate sufficiently from the original orientation [14].

The deformation field around the inclusion is heterogeneous and results in plastic strain gradients that play a role in recovery and recrystallisation. However, it is possible for such deformation to relax, for example by the dissipation of defects into the surroundings, especially when the inclusions are small in size. For large inclusions, features such as dislocation loops cannot be removed from the neighbourhood of the particles so relaxation processes become less effective. As a consequence, particles that are generally several micrometres in size act to stimulate recrystallisation. Therefore, fine dispersions, such as those associated with microalloyed steels, retard recrystallisation by pinning boundaries or dislocations, rather than accelerating it.

• Given the mechanism by which recrystallisation nucleates, i.e., grain boundary bowing or polygonisation, it is natural that the crystallographic character of boundaries in the deformed state will influence the development of crystallographic texture in the recrystallised form. It has, for example, been found that in recrystallisation experiments on ferritic bicrystals, an initial “у-fibre”2 texture results in recrystallised grains that are significantly rotated but still on the у-fibre of the orientation distribution function [14].

### Phenomenological treatment of recrystallisation

Thermomechanical processing is routine in the production of structural steels, with the aim of refining the austenite grain size and hence the ferrite grain size following transformation. But the rate of production is impressively large; the time spent within the austenite phase field during rolling may be less than two seconds during each rolling reduction, although the delay between rolling-passes may be of the order of two minutes. The structural changes during these short time scales are special and yet relatively simple to model empirically [15].

During hot deformation, the stress required to deform the steel is a function of the plastic strain (ep), the plastic-strain rate (ep) and temperature. That stress is a function /{ep} of plastic strain, as is well-understood from any tensile test of a steel. Zener and Holloman proposed [16, 17] that the effects of strain rate and temperature can be combined by writing a = /{e,Zk} with Zk, now known as the Zener-Hollomon parameter, defined as

where Q is an unspecified heat of activation since most rates are associated with an activated event. The material work hardens during hot-rolling but softens beyond a critical strain £* corresponding to the recrystallisation of austenite, with

where L0 is the austenite grain size prior to deformation and bj are empirical constants. Typical values of the empirical constants are Q = 312kJmol_l; bji = 6.97 x 10-4; Ьгъ = 0.3 when the grain size has units of micrometres, and Ьгл = 0.17 when the strain rate has units of reciprocal seconds [18].

If the strain £* is reached while the steel is still being rolled, then the process of change is known as dynamic recrystallisation. On the other hand, metadynamic recrystallisation is said to occur when recrystallisation follows immediately after a rolling pass when the strain retained in the austenite exceeds that needed to induce recrystallisation. The recrystallised austenite grain size will in general be smaller than the initial size, and grain growth during the interval between passes is inevitable, as illustrated in Figure 13.4 [19]. The full theory for recrystallisation and grain growth is not presented here because it tends to be alloy specific, but can be accessed from extensive literature on the subject [20,21].

Figure 13.4 (a) Influence of a single pass of rolling deformation to a strain £p = 0.3, on the austenite grain size and residual strain (0.3 multiplied by the fraction of unrecrystallised austenite) (b) Influence of four rolling- passes on the austenite grain size and residual strain. In each pass, recrystallisation is complete at points such as ‘a’, followed by grain growth in regions such as ‘b’. The units of ep are in s-1. Selected data from Bombac et al. [19].

The general features of controlled rolling are summarised in Figure 13.4. Really quite sophisticated process models now exist to treat the entire sequence of rolling [22, 23], microstructural development [24] and properties [25, 26], so much so that some of these are now used in the on-line control [27] of rolling mills using machine learning methods [28] to ensure product uniformity.

### Thermomechanical processing: limits to grain refinement

Grain size refinement using thermomechanical processing is an important method for improving both the strength and toughness of steels. It is useful therefore to consider the smallest ferrite grain size that can be achieved using this manufacturing method, by balancing the driving force for transformation from austenite to ferrite against the stored energy due to grain boundaries [29]:

which for equiaxed grains becomes

It follows that the smallest ferrite grain size that can be achieved is when all of is used up in creating a/a grain boundaries:

The term 2oy/Ly supplements the driving force the ferrite when it forms, eliminates the austenite grain boundaries. Obviously, a reduction in the austenite grain size should always lead to finer ferrite grains but the magnitude of the change depends also on |AGya|, i.e„ on the undercooling at which the у a transformation occurs. The austenite grain size becomes less important at large undercoolings.3

Figure 13.5 shows the ferrite grain size (E‘™n) as a function of the driving force using Equation 13.7 with C7« = 0.6 J m-2. Also illustrated are data from industrial processing. At large undercoolings, the size achieved is much bigger than expected theoretically. This is because of recalescence caused by the larger enthalpy of transformation at greater undercoolings, which heats the steel to higher than intended temperatures, thereby reducing ДGya. Once this is accounted for, the analysis indicates that it probably is not possible to obtain allotriomorphic ferrite grain-sizes much smaller than 1 pm using thermomechanical processing of the type used in mass production.

Figure 13.5 Plot of the logarithm of ferrite grain size versus the free energy change at Ari. The ideal curve represents the values of L™in. The points are experimental data; in some cases it is assumed that the grain size quoted in the literature corresponds to the mean lineal intercept. The sources of the data are quoted in [29].