Steady-State Grain Size in Dynamic Recrystallization of Minerals

where σ is the flow stress, μ is the shear modulus, b is the length of the Burgers vector, and K is a non-dimensional constant. The grain size exponent p ranges between 1 and 1.5 for most materials. Empirically determined σ–d relations of minerals have been used to estimate the stress states in the Earth’s interior. However, detailed studies of a Mg alloy (De Bresser et al., 1998) and NaCl (Ter Heege et al., 2005) revealed that K has a weak dependence on temperature. Derby & Ashby (1987) modeled the DRX processes of metals and predicted the temperature dependence of the recrystallized grain size, but they failed to account for the observed range of exponent p (Derby, 1992; Shimizu, 2011).


Introduction
Dynamic recrystallization (DRX) is a strain restoration and grain refinement mechanism that occurs in high-temperature dislocation creep of metals and minerals (Humphreys & Hatherly, 2004). Microstructures indicative of DRX are commonly observed in rock-forming minerals that have been subjected to natural deformation in the Earth's crust and mantle (Fig. 1).
Laboratory studies have revealed that the average size d of recrystallized grains approaches a steady-state value, which is determined by the applied stress and is independent of the initial grain size. Twiss (1977) proposed a stress-grain size relation of the following form: where σ is the flow stress, μ is the shear modulus, b is the length of the Burgers vector, and K is a non-dimensional constant. The grain size exponent p ranges between 1 and 1.5 for most materials. Empirically determined σ-d relations of minerals have been used to estimate the stress states in the Earth's interior. However, detailed studies of a Mg alloy (De Bresser et al., 1998) and NaCl (Ter Heege et al., 2005) revealed that K has a weak dependence on temperature. Derby & Ashby (1987) modeled the DRX processes of metals and predicted the temperature dependence of the recrystallized grain size, but they failed to account for the observed range of exponent p (Derby, 1992;Shimizu, 2011).
In this chapter, we focus on deformation and recrystallization processes in minerals and examine the effects of stress and temperature on the steady-state grain size.

Recrystallization mechanisms in minerals
DRX was first observed in hot deformation of cubic metals such as Cu, Ni, and austenitic iron. A simplified description of DRX in these metals is as follows. Strain-free new grains are usually formed by bulging of pre-existing grain boundaries and they grow at the expense of old grains to reduce the dislocation energy of the material (Sakai, 1989;Sakai & Jones, 1984).
As the dislocation density of the new grains increases, they cease to grow and new nucleation events occur at their margins. These processes repeat cyclically during dislocation creep.

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www.intechopen.com In contrast to the classical view of DRX described above, syndeformational recrystallization of minerals such as quartz, calcite, and olivine proceeds with progressive misorientation of subgrain boundaries (Poirier, 1985). Subgrain rotation (SGR) recrystallization also occurs in some metals such as Mg and Al alloys and is termed continuous DRX, whereas DRX in the original sense is currently referred to as discontinuous DRX (Humphreys & Hatherly, 2004). At low temperatures (T) and high strain rates (ε), SGR is localized at grain margins (Hirth & Tullis, 1992;Schmid et al., 1980); however, intracrystalline SGR becomes more important and grain boundary migration (GBM) occurs at high T and lowε (Hirth & Tullis, 1992;Rutter, 1995) (Fig. 2). Consequently, the recrystallized grain size is much larger than the subgrain size (Guillopé & Poirier, 1979;Karato et al., 1980).
For both discontinuous and continuous DRX, grain size reduction occurs at nucleation events, whereas strain-induced GBM leads to overall coarsening. The steady-state grain size is determined by the dynamic balance between nucleation and grain growth (Derby & Ashby, 1987).

Grain size distribution
In the σ-d relation (Eq. 1), the steady-state microstructure is represented by a single value of the 'average' grain size d, but dynamically recrystallized materials generally have wide grain size distributions. As a simplified model of DRX, Shimizu (1998a;1999;2003) considered following nucleation and growth processes and analyzed the evolution of the grain size distribution: 1. Nucleation occurs at a constant rate I per unit volume.
3. Each grain grows with a radial growth rateṘ. 4. Newly crystallized grains replace older grains.
In the steady state, the grain size has a nearly a log-normal distribution and many newly crystallized grains coexist with a few old grains in a certain population balance. The average grain size satisfies where a is a scaling factor; a = 1.14 for a 3D distribution and a = 1.12 for a distribution measured in a 2D section. Shimizu (1998b;2011) considered strain-induced grain growth forṘ (Sec. 4) and SGR nucleation for I (Sec. 5) and derived the σ-d relation for continuous DRX (Sec. 6). In Sec. 7, we revise the theoretical model to incorporate the influence of the surface-energy drag.

Dislocation energy
During high-T dislocation creep of minerals, dynamic recovery cooperates with continuous DRX and assists subgrain formation. Unrecovered microstructures such as tangled dislocations are rarely observed in recrystallized grains (Hirth & Tullis, 1992). Hence, the strain energy (E strain ) is given by a sum of the energies of isolated dislocations and sub-boundaries (E disl and E sub , respectively): Steady-State Grain Size in Dynamic Recrystallization of Minerals www.intechopen.com The free dislocation energy per unit volume is where ρ is the dislocation density and ζ is the dislocation line tension. When the internal stress around dislocations is equilibrated with the applied stress σ, the following equation holds (Nabarro, 1987): where α is a constant that depends on the configuration of the dislocation arrays. Hence, The dislocation line tension is given by (Hirth & Lothe, 1982) where r is the characteristic radius of the elastic field around the dislocation core and the constant β is typically in the range 3-4. The parameter χ depends on the dislocation configuration: where ν is Poisson's ratio. For a first-order approximation, we assume that all dislocations are edge dislocations. Considering that the elastic field around a dislocation is canceled by other dislocations at half the distance between them, r is scaled as Substituting Eqs. (6) and (9) into Eq. (7) yields Substituting Eq. (10) into Eq. (4) and using Eq. (6) again, we have

Sub-boundary energy
Consider nearly spherical subgrains with a diameter d ′ that occupy a deformed matrix (Fig.  2a). The number density of subgrains is and the area of subgrain boundaries per unit volume is The factor 1/2 is included because the area of each subgrain wall is counted twice. The energy of sub-boundaries in a unit volume of the material can thus be written as where γ is the sub-boundary energy per unit area.
The theory of dislocations gives where h is the mean dislocation spacing (Hirth & Lothe, 1982).

Subgrain size
We consider a recovery process in which free dislocations with a dislocation density ρ rearrange into sub-boundaries. Conservation of the total dislocation length during subgrain formation requires The right-hand side represents the length of dislocations in sub-boundaries. Using Eqs. (13) and (18), the above expression is modified to become Subgrains are formed if the total sub-boundary energy is smaller than the free dislocation energy (Twiss, 1977): The equality represents the critical state for the initiation of subgrain formation. From Eqs. (4) and (14), this condition can be written as Fig. 3. Schematic illustration of a tilt boundary with a misorientation angle θ, dislocation spacing h, and Burgers vector b.
Substituting Eqs. (6), (7), and (21)-(22) into the above expression for ρ, ζ, and γ, respectively, Eq. (28) becomes σ αμb Equating Eqs. (26) and (6), we have Then, Eq. (29) reduces to The stability limit of θ is derived as where e is the Napierian base. The equality gives the initial misorientation angle θ i : Applying θ i to θ of Eq. (30), the initial subgrain size d ′ i is obtained as Once the subgrain boundary is established, it functions as a dislocation sink because progressive subgrain misorientation is an energetically favorable process. We thus assume that the subgrain size is maintained during the subsequent misorientation. Substituting d ′ = d ′ i into Eq. (34), we obtain Eq. (23), where Using Eqs. (22) and (35), the full expression of Eq. (24) is obtained as

Growth kinetics
The kinetic law of grain growth is generally written aṡ where M is the mobility of the grain boundary and F is the driving force. M depends on T as  (11) and (36). The physical parameters of quartz are given as   (Kohlstedt & Weathers, 1980), μ = 4.2 × 10 4 MPa, and ν = 0.15 (Twiss, 1977). Because no data are available for β of quartz, we apply β = 3of ionic crystals (Hirth & Lothe, 1982).
where w is the boundary width, k is the Boltzmann constant, D gb is the diffusion coefficient at the grain boundary, D • gb is a constant, R is the gas constant, and Q gb is the activation energy for grain boundary diffusion.
In a single-phase material, grain growth occurs to reduce the bulk strain energy and the energy of grain surfaces. Hence, Eq. (37) is written aṡ where F strain and F sur f represent the driving forces due to strain energy and surface energy (grain boundary energy), respectively. The strain energy in dynamically recrystallized materials is not homogeneous. The strain energy of deformed grains is given by the sum of E disl in Eq. (11) and E sub in Eq. (36), whereas newly recrystallized grains are almost strain free. This difference in strain energy drives grain growth. Hence, With increasing strain, free dislocations multiply and excess dislocations rearrange into sub-boundaries. Then, θ increases and the sub-boundary energy exceeds the free dislocation energy. Fig. 4 shows the calculations for quartz. When the average misorientation angle reaches several degrees, the following approximation can be used instead of Eq. (3):

Nucleation rate
In SGR nucleation, the nuclei are approximately the same size as the original subgrains. Thus, the number of potential nucleation sites per unit volume of crystals is given by Eq. (12) for intracrystalline nucleation and for nucleation at grain margins (Fig. 2b). The nucleation rate is scaled as where τ c is the interval of nucleation events.
The subgrain becomes a nucleus when the misorientation angle θ exceeds a critical value θ c . The flux of dislocations that move toward the sub-boundary is given by ρu, where u is the climb velocity. The time required for dislocations to accumulate at the sub-boundary is equal to the nucleation cycle τ c . From Eq. (18), a critical nucleus has a dislocation spacing of h c = b/θ c ; hence, the number of dislocations per unit area of the boundary is 1/h c = θ c /b. Dividing this value by the flux ρu, the nucleation cycle is evaluated as The climb velocity of dislocations is given by (Hirth & Lothe, 1982) where Ω is the atomic volume, D v is the self-diffusion coefficient, and l is a length scale given by Using Eqs. (6) and (9), Eq. (47) can be rewritten as The temperature dependence of D v is expressed as where D • v is a constant and Q v is the activation energy for volume diffusion.

Scaling relation
Here, we neglect the surface energy term in Eq. (40) and assume for intracrystalline nucleation and for marginal nucleation (Shimizu, 1998b;. B is a non-dimensional constant given by Although σ is included in the right-hand side, the stress dependence of B is negligibly small. Using Eqs. (39) and (49), Eq. (53) can be re-expressed as where and Fig. 5. Schematic representation of grain size evolution due to (a) strain-energy-driven grain growth and (b) surface-energy drag.
As Q gb is generally smaller than Q c , the recrystallized grain size is predicted to have a weak positive dependence on T. The constant K in Eq.
(1) can now be written as a function of T:

Influence of surface energy
We now consider the influence of surface energy (grain boundary energy). In the case of surface-energy-driven grain coarsening in single-phase materials under static conditions (known as normal grain growth), large grains are energetically favorable and grow at the expense of small grains; the evolution of individual grain size has the opposite sense to that considered for DRX in Sec. 3 (Fig. 5). Therefore, when new grains grow by the strain-energy difference, the surface energy acts as a drag force.
In the theory of normal grain growth (Hillert, 1965), grain size evolution is described bẏ where R k andṘ k are respectively the radius and the growth rate of the k-th grain and c ∼ 1isa statistical factor. If R k is smaller (larger) than the mean radius R, the above expression becomes negative and the k-th grain shrinks (grows). By comparison with Eq. (37), the driving force for the growth of the k-th grain can be written as where d k is the diameter of the k-th grain. In the nucleation and growth processes in DRX, the influence of the surface-energy drag is largest for small nuclei. Thus, we introduce a modified factor c ′ and express the surface-energy-driven force in Eq. (40) as With this equation and Eq. (42), Eq. (40) can be approximated aṡ Using this equation, Eq. (56) can be modified as follows (the parameters p, m, and ΔQ remain the same).

Stress dependence of recrystallized grain size
In Fig. 6, p values of rock-forming minerals determined by triaxial or uniaxial or compression tests are plotted against the n-th power of dislocation creep flow laws (ε ∝ σ n ), which reflect the rate-controlling processes of dislocation creep; for climb-controlled creep, n is generally 3-5. The figure also shows the experimental result for a hexagonal Mg alloy (Magnox Al80), which was studied as a quartz analogue (De Bresser et al., 1998). The observed p values are almost independent of the power-law exponents and are well explained by the present model for continuous DRX.

Application to quartz
The theoretical model for the recrystallized grain size was applied to quartz using the equations presented in Sec. 6 2011). However, the previous model accounted only for strain energy; it neglected the effects of surface energy. Moreover, it turned out that the previous calculation involved a numerical error; when this error is corrected, the theoretical σ-d lines (Fig. 8 of ) shift to higher σ. Here, we recalculate the σ-d relation of quartz using the revised equations in Sec. 7.
In Fig. 7(b), the theoretical model is extended to the α-quartz stability field in which D v of oxygen in α-quartz (Farver & Yund, 1991a) is used and α-and β-quartz are assumed to have the same Q v /Q gb ratio. The recrystallized grain size of α-quartz is predicted to be d = 9.98 × 10 2 × σ −1.25 exp 12.4 kJ/mol RT ; α−quartz With decreasing temperature, the steady-state grain size shifts to higher stresses. If the empirical σ-d relation is directly applied to natural rocks that have deformed under low-T (≤ 400 • C) metamorphic conditions, the stress states will be considerably underestimated. recrystallized grain size at 1000-1100 • C after Stipp & Tullis (2003). Black dotted line: empirical d-σ relation across the temperature range of 700-1100 • C after Stipp & Tullis (2003). (b) Theoretically predicted σ-d relations for β-quartz (blue lines, 1000-600 • C) and α-quartz (red lines, 500-300 • C) using the intracrystalline nucleation model.

Summary
High-T dislocation creep of minerals is characterized by the occurrence of continuous DRX. The steady-state grain size is determined by the dynamic balance between SGR nucleation and grain growth by GBM. Surface energy acts as a drag force for strain-energy-driven GBM. The negative dependence of recrystallized grain size on stress is well explained by a theoretical model for continuous DRX. The theory also predicts a weak positive dependence of recrystallized grain size on temperature.