Phase-Field Modeling of Precipitation

In PanPhaseField, a multi-component multi-phase phase field method (MPFM) [1996Ste, 2011Guo, 2018Shi] is adopted to simulate precipitation kinetics. Elastic constants and stress-free transformation strain (SFTS) are used to calculate elastic energy of precipitate and particle morphology can be precisely predicted.

For a system with α =1,2,…,N phases and i=1,2,…,M components, its total free energy functional (F) is given as a sum of the interfacial energy density (f ^intf), the bulk chemical free energy density (f ^chem) and the elastic energy (Eel).

(1)

The order parameter field, , represents the local volume fraction of the phase with its concentration field described by the vector：

(2)

Concentration field of the i-th component is represented as . In MPFM, the order parameter fields follow the constraint:

(3)

From the order parameter and the phase composition , the local concentration can be determined using the mass conservation relation:

(4)

PanEngine/PanDataNet is coupled with MPFM through the chemical energy density f ^chem,

(5)

Free energy of each individual phase, , is calculated from PanEngine. For the given local composition and phase field , individual phase composition vectors are evaluated through KKS model [1999Kim].

To solve for the individual phase concentration, it is assumed that local diffusion potentials are same in all the phases, which is expressed mathematically [1999Kim] as

(6)

where α and β are the phase indices and i is the component index. For a given set of order parameter fields and composition fields , phase compositions are uniquely determined by the Eq. 4 and Eq. 6.

The interfacial free energy density is given by

(7)

where and are the gradient energy coefficients and potential energy humps across interface and they together determine the interfacial energy and boundary width .

The interfacial energy between two phases α and β is evaluated to be:

(8)

The elastic strain energy dominates particle morphology and variant selection at coarsening stage. Elastic strain energy functional, , is formulated according to Khachaturyan’s microscopic elasticity theory [2013Kha] and modified in the framework of MPFM [2006Ste]. The key inputs in the formulation of are the SFTS (stress-free transformation strain) of each phase and the elastic constants C_ijkl of each phase.

The SFTS is defined in form of a matrix:

Generally, matrix phase, for example Fcc, plays as reference phase of this SFTS matrix. When a cubic-to-cubic transformation from matrix to precipitate phase happens, e₁₁ = e₂₂ = e₃₃and e_ij = 0 for i≠ j. When a cubic -> tetragonal transformation occurs, the SFTS could be one of e₂₂ = e₃₃or e₁₁ = e₃₃ or e₁₁ = e₂₂ depending on the orientation variants of precipitation. Other symmetry breaking will introduce a much more complex form of SFTS, which is beyond discussion in this manual.

The coherency elastic strain energy is formulated following Khachaturyan’s microscopic elasticity theory [2013Kha] under the homogeneous modulus assumption when external applied stress is zero:

(9)

Where p and q are the index of two phases. n is a unit vector in the reciprocal space along the direction of k. is the Fourier transformation of the absolute value of the structural order parameter . The superscript asterisk in denotes the complex conjugate. is calculated as:

(10)

The phase fields and concentration fields are evolved in time through MPFM and Cahn-Hilliard dynamics respectively,

(11)

(12)

where is the chemical mobility [2004Che], is the interface mobility and is the number of phases that coexist locally. , where and are the chemical potentials of components k and n, respectively. Chemical mobility of a phase [2004Che] is given as

(13)

where M^l is the atomic mobility and c^lis the composition of l-th element. The Eq. 11 and Eq. 12 are solved concurrently.

In the precipitation model, there are several treatments related with kinetics of order parameters:

• Driving-force scaling method: Sometimes, when we model the precipitate growth, the driving force for transformation may become extremely large which may cause numerical instability. Hence, we use a scaling approach to reduce the driving force for phase transformation.

• Mobility of order parameter: For diffusion-controlled precipitation processes, the mobility of the order parameter ()should be large enough to ensure diffusion-controlled interface movement. But if is too large, we will observe the congruent transformation. Hence, we choose an appropriate inside PanPhaseField automatically to maintain this diffusion-controlled assumption.

• Interface width: The interface width in the KKS model is simply a numerical parameter and does not correspond to the real interface width. For numeric stability, we recommend an interface width of 5 grid points.

The pre-exponential term in Eq. 14 are: N_v, the nucleation site density, Z, the Zeldovich factor and β^*, the atomic attachment rate. t is the time, τ is the incubation time for nucleation, k_B the Boltzmann constant and T the temperature. The ΔG* is the nucleation barrier which is calculated according to interfacial energy, chemical driving force and elastic strain. For details of the nucleation rate calculation, please refer to Section Precipitation Nucleation.

The nucleation implanting algorithm in PanPhaseField follows a probabilistic Poisson seeding process at a probability of forming a critical nucleus [2019Shi]:

(15)

Where P(r,t) is the probability of forming a nucleus with critical radius in a volume ΔV and a time interval Δt and position r starting from time t.