Phase field modelling allotropic transformation of solid solution

Based on multiphase field conception and integrated with the idea of vectorvalued phase field, a phase field model for typical allotropic transformation of solid solution is proposed. The model takes the non-uniform distribution of grain boundaries of parent phase and crystal orientation into account in proper way, as being illustrated by the simulation of austenite to ferrite transformation in low carbon steel. It is found that the misorientation dependent grain boundary mobility shows strong influence on the formation of ferrite morphology comparing with the weak effect exerted by misorientation dependent grain boundary energy. The evolution of various types of grain boundaries are quantitatively characterized in terms of its respective grain boundary energy dissipation. The simulated ferrite fraction agrees well with the expectation from phase diagram, which verifies this model.


Introduction
The microstructure and the texture information from the allotropic transformation of solid solution, for example γ→α transformation in iron-carbon alloy or β→α transformation in titanium alloy, plays important role in its heat treatment process and thermomechanical process [Delaey (1991)]. The transformation is complicated in view of decomposition process, grain growth and texture evolution. As a powerful tool, phase field is capable of modelling complex microstructure efficiently such as grain growth  ;Jamshidian, Zi and Rabczuk (2014); Moelans, Blanpain and Wollants (2008); Steinbach (2009)] and crack propagation [Amiri, Millán, Shen et al. (2014); ; ; Areias, Reinoso, Camanho et al. (2018);Msekh, Cuong, Zi et al. (2018)]. Although great effort has been made all along by phase field modelling the transformation [Militzer (2011)]. Proper phase field model is absent for the moment which is able to consider both the effects of grain boundary (GB) in parent phase and crystal orientation (referred as texture) on the formation of new phase. For example, as a typical allotropic transformation of solid solution, the microstructure evolution during the transformation of austenite (γ) to ferrite (α) has previously been modelled extensively using the phase field approach. The study of γ→α transformation mechanism by modelling is carried out in the pioneering work of Yeon et al. [Yeon, Cha and Yoon (2001)] and Loginova et al. [Loginova, Odqvist, Amberg et al. (2003)]. Huang modelled the transformation at a relatively large space scale by inputting the prior initial austenite GB as the selected nucleation site [Huang, Browne and McFadden (2006) ], which considers the austenite phase as spatially uniform in fact. Nucleation of new phase and spatial non-uniform of parent phase is reproduced in Militzer et al.'s work by a series of 2D and 3D simulation of γ→α transformation [MecozziG, Sietsma and van der Zwaag (2005)；Mecozzi, Sietsma, ; Mecozzi, Sietsma and van der Zwaag (2006);Militzer, Mecozzi, Sietsma et al. (2006);Mecozzi, Militzer, Sietsma et al. (2008)]. These fruitful simulations utilize fixed values of parameters such as interfacial energy and mobility, so that the growth morphology tends to be texture irrelevant. To cope with this, Yamanaka et al. artificially set the interfacial mobility to be a function of crystallographic misorientation depending on grain crystal orientation [Yamanaka, Takaki and Tomita (2008)]. It can be observed that the exist models either ignore or introduce these aspects artificially by setting up information in advance, assuming that the evolution of grain boundary in parent phase or crystal orientation of new phase during phase transformation and grain growth is negligible. It should be admitted that these assumptions are reasonable to some extent from the viewpoint of statistics when comparing with experimental results. While for a restrict study, comprehensive information is required in primary model because statistics is not able to reflect every details of transformation which is important in phase transformation kinetics. At the mention of kinetics, phase field model is good at microstructure and texture evolution as an outstanding methodology [Moelans, Blanpain and Wollants (2008)]. The purpose of the present work is to develop a phase field method which is able to account for texture evolution, grain growth during phase transformation and hence to study the allotropic transformation kinetics of solid solution, including the effect of parent phase and texture on the morphological development of new phase. The proposed model and its related parameters are presented in Sections 2 and 3. Simulation and discussion is shown in Section 4.

Phase field model for typical allotropic transformation of solid solution
Given that strain fields can be neglected, modelling solid-state transformation of austenite to ferrite (γ→α) requires a consideration of the following aspects [Delaey (1991)]:  The grain boundaries of parent austenite phase as matrix, i.e., spatially non-uniform;  The formation of ferrite phase and its subsequent grain growth with hard impingement;  Integration of phase transformation, grain growth and crystal orientation evolution. Considering a system with austenite grains coexisting with ferrite grains, the system free energy which satisfies these above requirements is proposed as follows: Here the existence of each austenite grain at location r at time t is marked by ϕi(r, t)=1(i=1, 2…n-1), otherwise ϕi(r, t)=0; The m ferrite grains are treated as an ensemble by phase field variable ϕn(r, t)=1, otherwise ϕn(r, t)=0. Thereby the traditional multiphase field concept [Steinbach, Pezzolla and Nestler (1996)] is based and the vector-valued phase field is introduced as follows. Each of m ferrite grains is described by where θ(r, t) is the normalized crystalline orientation field in 2D [Kobayashi, Warren and Carte (1998)]. As similar as the phase field, it takes constant values inside each grain and changes gradually across boundaries, composing a diffusive interface. ∇θ represents the inhomogeneity of orientation in space. The system free energy is therefore independent of the frame from which the crystal orientation is measured, making it rotation invariant and independent of grain crystal orientation. The linear term |∇θ| ensures the stability of the ferrite grain boundaries and a quadratic term |∇θ| 2 describes their motion [Kobayashi, Warren and Carte (1998)]. It is implicitly assumed in 2D that the boundaries/interfaces properties are independent of inclination which is another degree to describe anisotropy in 3D, therefore boundaries are isotropic and their properties is only a function of misorientation [Kobayashi, Warren and Carte (1998)]. G i (ci,T) is the homogeneous free energy density of grain i with composition ci at a temperature T. Vm is the molar volume. εij is the gradient coefficient and Wij is the double well potential height between grains. H and εθ specify the coupling strength between ϕn andθ. g(ϕn) and h(ϕn) are required to be monotonic as g(ϕn)=h(ϕn)=ϕn 2 . λL is Lagrange multiplier accounting for the limiting 1 n i i ϕ = ∑ . As a conclusion, Eq. (1) extends the system free energy of typical multiphase field [Steinbach, Pezzolla and Nestler (1996)] through introducing a vector-valued phase field [Kobayashi, Warren and Carte (1998)]. With this above system free energy, the time evolution of the non-conversed phase and orientation fields obeys the Ginzburg-Landau equation, whereas that of the conserved composition field follows the Cahn-Hilliard equation. The governing equations are as follows: where Mij is the phase field mobility, Mθ the orientation field mobility, D the diffusivity and sij the step function which specifies grain boundaries. Following Kim et al. [Kim, Kim, Suzuki et al. (2004)], the interface field method [Steinbach and Pezzolla (1999)] is used for the derivation of phase field equations (Eq. (2)) and the assumption of local equal interfacial chemical potential [Kim, Kim, Suzuki et al. (2004)] is employed to obtain interfacial compositions of coexisting phases. Comparing with traditional multiphase field model, the number of phase field equations is reduced from n+m-1 to n, which indicates equations number is independent of the number of ferrite grains. Though the calculation of orientation field is included, the computation load can be expected to be decreased by the reduction of the whole number of phase field equations.

Model parameters and numerical details
The gradient coefficient εij and potential well height Wij are related to grain boundary energy σij and interface width δ by Kim et al. [Kim, Kim, Suzuki et al. (2004) Phase field mobility Mij(or Mϕ) is related to grain boundary mobility M by Kim et al. [Kim, Kim, Suzuki et al. (2004)].
The orientation field mobility Mθ is chosen to suppress the rotation of the ferrite grains. The orientation field parameters εθ and H are related to εij and Wij [Kobayashi, Warren and Carte (1998)]. The thermodynamic data for the Fe-C system are listed in the Appendix. The temperature is set at 1050 K and c0=0.005 mole fraction; the equilibrium composition of ferrite cα=6.5739×10 -3 and austenite cγ =2.1056×10 -2 are calculated from the thermodynamic data. Tab. 1. lists the simulation parameters where Di(i=α,γ) is the diffusivity and σij(i,j=α, γ) is the boundary energy per unit area. The second-order explicit finite difference method on uniform grids is employed in calculating phase fields and composition field. Due to the singularity of orientation field equation, a fully implicit scheme is adopted. Six grids are assumed in the interface and grid size dx=2×10 -7 m is used. Unless otherwise stated, the size of the computational domain is 300×300 grids with periodic boundary conditions and a typical austenite grain size as 20 μm.
Figs. 1(a) and 1(b) show the simulated orientation field with misorientation dependent interface energy and with misorientation dependent mobility, respectively. According to Eq. 8, the α/γ1 interface is the high angle grain boundary (HAGB) while α/γ2 interface being the low angle grain boundary (LAGB). Simulation show that ferrite grows faster along the grain γ/γ boundary than its growth into the austenite in both cases, showing preference in locality. For case (a), little difference in growth morphology can be distinguished between HAGB and LAGB. This can be ascribed to that the chemical driving force due to transformation is prevailing over the free energy stored in interfaces, so that capillary effect is not an overwhelming priority. For case (b), a distinct difference in growth morphology is observed, showing the constrained ferrite growth into the austenite grain (LAGB) with the lower mobility. This observation can be ascribed to the significant difference in mobility will directly affect the magnitude of driving force according to Eq. (2), therefore the prevailing of chemical driving force is indirectly presented. The observation suggests that the growth morphology of ferrite is sensitive to an orientation dependent interfacial mobility rather than an orientation dependent interfacial energy per se. Although these two parameters are in principle related, they can be independently controlled by impurities segregation [Christian (2002)]. It might in fact be expected that interfacial energy has no overriding influence on growth morphology at a stage if the free energy change due to phase transformation, i.e., the chemical driving force. Which is opposed to grain growth where the sole driving force is the minimisation of total grain boundary energy per unit volume.

Phase transformation kinetics-misorientation independent mobility
Due to the faint effect of misorientation dependent interface energy on ferrite growth, following simulations assume that all boundary energies are misorientation independent with the effect of crystallography being manifested through a misorientation dependent mobility alone. A Voronoi diagram is used to create a polycrystalline aggregate of austenite grains with a distribution of crystal orientation as marked on Fig. 2(a). Since this construction does not lead to mechanical equilibrium at triple junctions. It is necessary to relax the phase field describing the austenite grain structure (in the absence of composition or structure heterogeneities) in order to permit the equilibrium to be established, as shown in Fig. 2(b). Base on above austenite grains configuration, assuming nucleation of ferrite occurs preferentially at triple junctions (corners), six ferrite particles with orientations α1(0.71), α2(0.40), α3(0.50), α4(0.88), α5(0.26) and α6(0.60) are introduced in order to model its growth with misorientation-independent mobility. The microstructure evolution and the crystal orientation distribution of transformed microstructure are shown in Fig. 3(a) and Fig. 3(b), respectively. Observation of the composition field in Fig. 3(c) reveals that soft impingement occurs at an early stage of transformation. As transformation progresses, the growth of ferrite is slowed down at localities where significant soft impingement 〈1 ensues due to the building-up of carbon in the adjacent austenite. Hard impingement eventually takes places between different grains of ferrite. Since grain boundary energy density is defined in prior by Eq. (1), the corresponding grain boundary energy associated with each type of boundary (γ/γ, γ/α and α/α) and its gross value can be calculated as follows: The characters of phase transformation kinetics can be revealed in detail by observation of energy dissipation of each kind of GB/interface. As shown in Fig. 4, the γ/γ grain boundary energy decreases monotonically since γ/γ grain boundaries continuously consumed during phase transformation. The α/γ grain boundary energy decreases initially for a very short duration due to the relaxation of initial state, then it increases monotonically as the fraction of ferrite increases. The ultimate reduction results from the replacement of α/γ by α/α because of ferrite grains get in touch with each other as transformation goes. The contribution from α/α grain boundaries is small because the isothermal transformation (1050 K) places restriction on final ferrite fraction, resulting in only part of ferrite grains interaction, i.e., hard impingement. The overall process can be categorised into three stages (Fig. 4) according to total grain boundary energy dissipation: The first stage involves the rapid preferential growth of ferrite at available parent austenite triple junction's area, resulting in a quick and considerable GB energy reduction. The subsequent second stage indicates that the rapid growth of ferrite into austenite grain surpasses the consumption of available γ/γ boundary. The third stage is where the transformation slows down and equilibrium is finally approached. The obscure fluctuation characters α/α GB energy in Fig. 4 can be revealed apparently at length when it is scaled up by curve a in Fig. 5 as transformation goes, which is expected being caused by the sharpness of orientation field. The onset of hard impingement leads to a rapid rise in its GB energy at about 20 s (the insert map in Fig. 5). A reduction follows the climax is caused by curvature driven coarsening of ferrite grains beyond about 38 s. Meanwhile, the transformation gradually comes to the end and fraction of ferrite come close to equilibrium value, as shown in curve b in Fig. 5.

Phase transformation kinetics-misorientation dependent mobility
The simulation of ferrite growth with a misorientation-dependent mobility is presented in Fig. 6. Comparing with Fig. 3, the obvious distinction for transformed ferrite morphology can be observed. Typical transformation characters are exemplified in Fig. 6(a), in which one ferrite nucleus and three austenite grains contacts each other being marked as α1(0.71), γ1(0.33), γ3(0.22), γ6(0.66). According to Eq. (2), Mα1/γ6=6.63e-3*Mα1/γ1, therefore the ferrite growth into the γ6 grain is severely limited by this low mobility, compared with its considerable penetration into γ1 and γ3. The more limited growth leads to the higher degree of solute diffusion, which is indicated in the composition field by Fig. 6(c) with obscure pile-up of solute in austenite at the front of α/γ6 interface. Fig. 6(b) shows the distribution of crystal orientations. The comparison of ferrite fraction during isothermal transformation is illustrated in Fig. 7 for the case of misorientation independent mobility and misorientation dependent mobility. It can be observed that the fraction in both cases gradually converges to the equilibrium value (0.7871). However, because of the overall averaged smaller value of misorientation dependent mobility than that of misorientation independent mobility, the longer time is needed to achieve equilibrium ferrite fraction.

Conclusions
To simulate the isothermal phase transformation of ferrite from austenite in a Fe-C alloy, a phase field model has been proposed by combining a vector-value phase field and the traditional multiphase field. By this model, the effect of crystal orientation on transformation is investigated by introducing the misorientation dependent variables including interfacial energy and GB mobility. Simulation show that the morphology of early formed ferrite is locally dependent on the GB distribution of parent austenite phase by the preferentially growth of ferrite along austenite GB. Meanwhile, misorientation dependent interface energy influence the evolution of the microstructure weakly in case of identical mobility for all the ferrite-austenite interfaces; a misorientation-dependent interfacial mobility on the other hand, has a major effect on ferrite morphology because the significant difference in mobility will directly affect the magnitude of driving force, especially when the chemical driving force is overwhelming comparing with capillary effect. According to quantitative analysis of total grain boundary energy dissipation, the transformation shows the character of three stages: The rapid preferential growth of ferrite at available parent austenite triple junction's area occurs firstly. The rapid growth of ferrite into austenite grain which surpasses the consumption of available γ/γ boundary comes into being the second stage. Finally, the transformation slows down and equilibrium is approached. Meanwhile, misorientation-dependent growth leads to a lower rate of transformation since a fraction of the transforming fronts always have low overall averaged mobility, together with the obvious distinction for transformed ferrite morphology from misorientation-independent growth. Appendix A. Thermo-dynamic data of Fe-C binary system [Gustafson (1985)] For ferrite phase: