In Situ Observation of Changing Crystal Orientations During Austenite Grain Coarsening

Abstract

Understanding the underlying mechanisms of grain coarsening is important to control the properties of metals, which strongly depend on the microstructure that forms during the production process or use at high temperature. Grain coarsening of austenite at 1,273 K in a binary Fe-2 wt% Mn alloy was studied using synchrotron radiation. The evolution of volume, average crystallographic orientation and mosaicity of more than 2,000 individual austenite grains was tracked during annealing. It was found that there exists an approximately linear relationship between grain size and mosaicity, which means that orientation gradients are present in the grains. The orientation gradients remain constant during coarsening and consequently the character of grain boundaries changes during coarsening, affecting the coarsening rate. Furthermore, changes in the average orientation of grains during coarsening were observed. The changes could be understood by taking the observed orientation gradients and anisotropic movement of grain boundaries into account. Five basic modes of grain coarsening were deduced from the measurements which include: anisotropic (I) and isotropic (II) growth (or shrinkage); movement of grain boundaries resulting in no change in volume but change in shape (III), movement of grain boundaries resulting in no change in volume and mosaicity, but change in crystallographic orientation (IV); and no movement of grain boundaries (V).

Appendix

1.1 Equation of Volume of a Grain

The diffracted intensity I _g per unit time from a single grain, rotated through the Bragg condition in order to illuminate the whole grain, is given by the following expression for the kinematic approximation (4, 5):

$$ I_{grain} = \frac{{I_{0} }}{\Delta \theta }\left( {\frac{{\mu_{0} }}{4\pi }} \right)^{2} \frac{{e^{4} }}{{m^{2} }}\frac{{\lambda^{3} F_{hkl}^{2} }}{{V_{c}^{2} }}V_{g} L_{g} P\exp ( - 2M) $$

(S(vi))

where I ₀ is the incident intensity of photons, F _hkl is the structure factor of the hkl-reflection, λ is the photon wavelength, V _g is the volume of the grain, Δθ is the change in scattering vector over which the grain is in reflection (Eq. S(ii)), V _c is the volume of the unit cell, $P = \left( {1 + \cos^{2} 2\theta } \right)/2$ is the polarization factor, and $L_{g} = 1/\sin 2\theta$ is the Lorentz factor, where 2θ is the scattering angle. The Debye-Waller factor exp(−2 M) accounts for the thermal vibrations of atoms, with

$$ M = \frac{{6h^{2} T}}{{mk_{B} \varTheta^{2} }}\left[ {\phi (x) + \frac{x}{4}} \right]\left( {\frac{\sin \theta }{\lambda }} \right)^{2} $$

(S(vii))

where h is the Planck constant, m is the mass of the vibrating atom, k _B is the Boltzmann constant, Θ is the Debye temperature, x = Θ/T is the relative temperature, T is the temperature and

$$ \phi (x) = \frac{1}{x}\int\limits_{0}^{x} {\frac{\xi }{\exp (\xi ) - 1}d\xi } $$

(S(viii))

The integrated intensity I _p per unit time of a hkl-diffraction ring of a polycrystalline material (often termed as powder in diffraction) with randomly oriented grains is given by (4, 5)

$$ I_{powder} = \varPhi_{0} \left( {\frac{{\mu_{0} }}{4\pi }} \right)^{2} \frac{{e^{4} }}{{m^{2} }}\frac{{\lambda^{3} m_{hkl} F_{hkl}^{2} }}{{V_{c}^{2} }}V_{gauge} L_{p} P\exp \left( { - 2M} \right) $$

(S(ix))

where m _hkl is the multiplicity factor of the hkl-ring and V _gauge is the volume of the diffracting phase. The Lorentz factor for a powder is given by L _p  = 1/(4sinθ).

The volume of an individual grain is calculated from the measured grain intensity I _g normalized by the powder intensity I _p of the hkl-ring in which the reflection from the individual grain appeared. In case the diffraction spot appears in more than one diffraction pattern, the intensity from the grain is divided by a factor k, equal to the number of diffraction patterns in which the spot is present. Combining Eqs. S(vi)) and S(ix) and introducing k gives Eq. S(i)

$$ V_{g} = \frac{1}{2}m_{hkl} \cos \left( \theta \right)V_{gauge} \frac{{I_{g} }}{{kI_{p} }}\Delta \theta $$

(S(i))

Equation S(i) is similar to the equation for volume of a grain used by Lauridsen et al. (6) and Offerman et al. (7). However, the following corrections have been made-

• 1. In the current analysis, diffraction spots distributed in more than one diffraction pattern were used for the volume calculation. Thus, the additional factor k is used.

• 2. The expression for Δθ used by Lauridsen et al. and Offerman et al. is $\Delta \theta = \Delta \omega \cdot \left| {\sin \left( \eta \right)} \right|$. However, this is an approximation which does not hold for low values of η. In the current analysis, Eq. S(ii) is used which is the exact form of the expression for Δθ. From Fig. S6, the expression for Δθ can be calculated as follows:

The projection along the x-axis of the vector (x, y, z) of length r in the direction of the plane normal before rotation is given as

$$ x = - r\sin \theta $$

After rotation, the vector is given as

$$ x' = - r\sin (\theta + \Delta \theta ) $$

(S(x))

Also, applying the rotation transform on vector r′ gives

$$ x' = - r\sin \theta \cos \Delta \omega - r\sin \eta \sin \Delta \omega \cos \theta $$

(S(xi))

Combining Eqs. S(x) and S(xi) gives Eq. S(ii)

$$ \Delta \theta = \sin^{ - 1} \left( {\sin \left( \theta \right)\cos \left( {\Delta \omega } \right) + \cos \left( \theta \right)\sin \left| \eta \right|\sin \left( {\Delta \omega } \right)} \right) - \theta $$

(S(ii))