The phase-field method was employed to model the evolution of the microstructure. This method has been used widely during the last two decades to simulate the microstructural evolution of materials.[
7,
8] The advantage of the phase-field method is that there is no need to track the interface, unlike the classical sharp interface modeling methods. An order parameter is introduced that varies smoothly between two phases, and thus the interface is part of the solution in the phase-field method.
Our simulations were performed using the commercially available phase-field modeling software MICRESS (version 6.400, Access e.V., Aachen, Germany). MICRESS is based on the multiphase-field approach.[
9,
10] The multiphase-field theory describes the evolution of multiple phase-field parameters
\( \phi_{\alpha = 1, 2, \ldots , v} = (\vec{x},t) \) (with the constraint
\( \mathop \sum \nolimits_{\alpha = 1}^{v} \phi_{\alpha } = 1 \)) in space and time, which represent the spatial distribution of multiple phases with different thermodynamic properties and/or multiple grains with different orientations. The phase-field parameter,
\( \phi_{\alpha } \) takes a value of 1 if phase
α is present locally and a value of 0 if the phase is not present locally. At the interface of the phase
α,
\( \phi_{\alpha } \) will vary smoothly from 0 to 1 over the interface thickness (
η). The time evolution of
\( \phi_{\alpha } \) is calculated using the free energy functional,
F, which integrates the density functional,
f, over the domain
\( {{\varOmega }} \).
$$ F\left( {\{ \phi_{\alpha } \} , \{ \vec{C}_{\alpha } \} } \right) = \int\limits_{{{\varOmega }}} {f(\{ \phi_{\alpha } \} , \{ \vec{C}_{\alpha } \} )} , $$
(1)
where the brackets, {}, represent all phases of
α, and not an individual
α. The density functional,
f, depends on the interface energy density,
\( f^{\text{int}} \), and chemical free energy,
\( f^{\text{chem}} \), and thus it can be written as follows:
$$ f = f^{\text{int}} \{ \{ \phi_{\alpha } \} + f^{\text{chem}} \{ \{ \phi_{\alpha } \} , \{ \vec{C}_{\alpha } \} \} , $$
(2)
$$ f = \mathop \sum \limits_{\alpha = 1}^{v} \mathop \sum \limits_{\beta \ne \alpha }^{v} \frac{{4\sigma_{\alpha \beta }^{0} a_{\alpha \beta }^{\sigma } }}{v\eta }\left( { - \frac{{\eta^{2} }}{{\pi^{2} }}\nabla \phi_{\alpha } \nabla \phi_{\beta } + \phi_{\alpha } \phi_{\beta } } \right) + \mathop \sum \limits_{\alpha = 1}^{v} \phi_{\alpha } f_{\alpha } (\vec{C}_{\alpha } ), $$
(3)
where
\( \sigma_{\alpha \beta }^{0} \) represents the interfacial energy of the interface between
α and
β.
\( v \) is the total number of local coexisting phases. The term
\( a_{\alpha \beta }^{\sigma } \) represents the anisotropy function for the interfacial stiffness.[
11] In 2D, for cubic crystal systems, this function takes the form
\( a_{\alpha \beta }^{\sigma } = 1 - \delta_{\sigma } \cos (4\theta ) \).[
12]
The multiphase-field equation defining the time evolution of
\( \phi_{\alpha } = (\vec{x},t) \) in multiple phase transformations is derived by minimizing the total free energy,
F, according to a relaxation principle.
$$ \dot{\phi }_{\alpha } = \mathop \sum \limits_{\beta \ne \alpha }^{v} M_{\alpha \beta } a_{\alpha \beta }^{M} \left( {\frac{\delta F}{{\delta \phi_{\beta } }} - \frac{\delta F}{{\delta \phi_{\alpha } }}} \right) $$
(4)
The general version of the evolution equation including the anisotropy can be written as follows.
$$ \dot{\phi }_{\alpha } = \mathop \sum \limits_{\beta \ne \alpha }^{v} M_{\alpha \beta } a_{\alpha \beta }^{M} \left[ {b_{\alpha \beta } {{\Delta G}}_{\alpha \beta } - \sigma_{\alpha \beta }^{0} a_{\alpha \beta }^{\sigma } K_{\alpha \beta }^{a} + \mathop \sum \limits_{\gamma \ne \beta \ne \alpha }^{n} J_{\alpha \beta \gamma } } \right] $$
(5)
$$ b_{\alpha \beta } = \frac{\pi }{\eta }\left( {\phi_{\alpha } + \phi_{\beta } } \right)\left( {\sqrt {\phi_{\alpha } \phi_{\beta } } } \right) $$
(6)
$$ \begin{aligned} K_{{\alpha \beta }}^{a} = & \,\frac{2}{v}\left\{ {\frac{{\pi ^{2} }}{{2\eta ^{2} }}\left( {\phi _{\beta } - \phi _{\alpha } } \right) + \frac{1}{2}\left( {\nabla ^{2} \phi _{\beta } - \nabla ^{2} \phi _{\alpha } } \right)} \right. \\ & \,+ \frac{1}{{a_{{\alpha \beta }}^{\sigma } }}\sum\limits_{{i = 1}}^{3} {\nabla _{i} } \left[ {\left( {\frac{{\partial a_{{\alpha \beta }}^{\sigma } }}{{\partial \nabla _{i} \phi _{\beta } }} - \frac{{\partial a_{{\alpha \beta }}^{\sigma } }}{{\partial \nabla _{i} \phi _{\alpha } }}} \right)\left( {\frac{{\pi ^{2} }}{{2\eta ^{2} }}\left( {\phi _{\alpha } \phi _{\beta } } \right) - \frac{1}{2}\left( {\nabla \phi _{\alpha } \nabla \phi _{\beta } } \right)} \right)} \right] \\ & \left. { - \frac{1}{{a_{{\alpha \beta }}^{\sigma } }}\nabla a_{{\alpha \beta }}^{\sigma } \left( {\nabla \phi _{\beta } - \nabla \phi _{\alpha } } \right)} \right\} \\ \end{aligned} $$
(7)
$$ \begin{aligned} J_{{\alpha \beta \gamma }} = \frac{2}{\nu} & \left\{ {\frac{1}{2}\left( {\sigma _{{\beta \gamma }}^{0} a_{{\beta \gamma }}^{\sigma } - \sigma _{{\alpha \gamma }}^{0} a_{{\alpha \gamma }}^{\sigma } } \right)\left( {\frac{{\pi ^{2} }}{{\eta ^{2} }}\phi _{\gamma } + \nabla ^{2} \phi _{\gamma } } \right) + \sigma _{{\alpha \gamma }}^{0} \sum\limits_{{i = 1}}^{3} {\nabla _{i} } \left[ {\left( {\frac{{\partial a_{{\alpha \gamma }}^{\sigma } }}{{\partial \nabla _{i} \phi _{\alpha } }}} \right)\left( {\frac{{\pi ^{2} }}{{2\eta ^{2} }}\left( {\phi _{\alpha } \phi _{\gamma } } \right) - \frac{1}{2}\left( {\nabla \phi _{\alpha } \nabla \phi _{\gamma } } \right)} \right)} \right]} \right. \\ & \,- \sigma _{{\beta \gamma }}^{0} \sum\limits_{{i = 1}}^{3} {\nabla _{i} } \left[ {\left( {\frac{{\partial a_{{\beta \gamma }}^{\sigma } }}{{\partial \nabla _{i} \phi _{\beta } }}} \right)\left( {\frac{{\pi ^{2} }}{{2\eta ^{2} }}\left( {\phi _{\beta } \phi _{\gamma } } \right) - \frac{1}{2}\left( {\nabla \phi _{\beta } \nabla \phi _{\gamma } } \right)} \right)} \right] \\ & \left. { + \frac{1}{2}\left( {\sigma _{{\beta \gamma }}^{0} \nabla a_{{\beta \gamma }}^{\sigma } - \sigma _{{\alpha \gamma }}^{0} \nabla a_{{\alpha \gamma }}^{\sigma } } \right)\nabla \phi _{\gamma } } \right\}, \\ \end{aligned} $$
(8)
where
\( K_{\alpha \beta }^{a} \) is related to the local curvature of the interface and
\( J_{\alpha \beta \gamma } \) relates to the third-order junction forces.
However, more simplified version of the
\( J_{\alpha \beta \gamma } \) term is implemented in MCRESS neglecting the higher order terms as follows.
$$ J_{\alpha \beta \gamma } = \frac{2}{v}\left\{ {\frac{1}{2}\left( {\sigma_{\beta \gamma }^{0} a_{\beta \gamma }^{\sigma } - \sigma_{\alpha \gamma }^{0} a_{\alpha \gamma }^{\sigma } } \right)\left( {\frac{{\pi^{2} }}{{\eta^{2} }}\phi_{\gamma } + \nabla^{2} \phi_{\gamma } } \right)} \right\}. $$
(9)
The interface motion depends on the curvature contribution,
\( (\sigma_{\alpha \beta } K_{\alpha \beta } ) \), but also on the thermodynamic driving force,
\( \Delta G_{\alpha \beta } \left( {\vec{C},T} \right) \). This driving force depends on the temperature,
T, and the local multicomponent composition,
\( \vec{C} \), which couples the phase-field equation to the multiphase diffusion equations:
$$ \dot{\vec{C}} = \nabla \mathop \sum \limits_{\alpha = 1}^{v} \phi_{\alpha } \vec{D}_{\alpha } \nabla \vec{C}_{\alpha } $$
(10)
$$ \vec{C} = \mathop \sum \limits_{\alpha = 1}^{v} \phi_{\alpha } \vec{C}_{\alpha } , $$
(11)
where
\( \vec{D}_{\alpha } \) represents the multicomponent diffusion coefficient matrix for the phase
α.
\( \Delta G_{\alpha \beta } \left( {\vec{C},T} \right) \) and
\( \vec{D}_{\alpha } \) are calculated by direct coupling to the thermodynamic (TCNI8) and mobility (MOBNI4) databases
via the TQ-interface in Thermo-Calc Software.[
13] The driving force,
\( \Delta G_{\alpha \beta } \left( {\vec{C},T} \right) \), is calculated based on the quasi-equilibrium approach with the combination of mass balance condition. For detail information, reader is advised to refer.[
10,
11]