Multi-Phase-Field Approach

Abstract

In this chapter the extension of a phase-field model for two phases to multiple phases is presented. This relates to the treatment of triple lines and junctions between several phases, or grains in a multicrystalline structure. The conservation constraint of the sum of all fields in one material point is realized using a Lagrange formalism. The free energy functional is expanded in pairs of phases, as well as the equation of motion of individual phase fields in dependence on all other fields. As example coarsening and texture evolution in a multi grain structure with anisotropic interface energy is presented.

1 Phase-Field Functional for Multiple Phases

Technical materials have multiple phases. In a CALPHAD (CALculation of PHAse Diagrams) database for iron alloys and steel, there are Gibbs-energy functions of several hundreds of different physical phases tabulated, and these are dependent on the crystal structure, composition, temperature, and pressure. According to Gibbs’ phase rule, there can, however, only be $n=2+n_{{ }_{\mathrm {c}}}$ stable phases in equilibrium, where $n_{{ }_{\mathrm {c}}}$ is the number of components of the alloy composition. For a steel, we may put $n_{{ }_{\mathrm {c}}}=10$, i.e., 12 different phases could be stable in equilibrium. Since, however, a steel is rarely in thermodynamic equilibrium but is more commonly in a kinetically stabilized off-equilibrium state with a locally varying composition, you may find 20 or more individual phases in a steel sample. In phase-field theory, we also attribute different grains—i.e., crystallites with the same crystallographic phase but a different orientation in space—to different phase fields. We do this to identify the interfaces between these grains, which will evolve in time (see Chap. 3). We can then have thousands of phase fields. How do we tackle this?

For completeness, there are two fundamentally different approaches for this “multi-grain” problem. The microscopic, or physical order-parameter models, use a free-energy landscape. Generally speaking, this is a special potential function with different minima that are dependent, for example, on the orientation of the crystallites [1, 11].

Another approach is the introduction of orientation as an order parameter [5, 6]. These approaches are generally considered as elegant from a theoretical point of view, but they are difficult to apply to practical problems. The reader should build his/her own opinion on this.

The third approach, which is more “brute force,” is to address each crystallite, regardless of its phase, by its own phase field. We call this the “multi-phase-field approach” [12, 13]. Also popular is the so-called “vector order parameter” model by Fan and Chen [2]. Both models consider a set of N phase fields ϕ_α (α = 1…N) that can be attributed to different properties such as orientation, composition, crystal structure etc. We will describe here only the multi-phase-field approach [12, 13]. We distinguish three cases, as depicted in Fig. 6.1:

• Bulk: only one phase field has the value of 1, all other fields have the value of 0.

• Dual interface: two phase fields have values between 0 and 1, all other fields are 0.

• Junctions: we define a number $\tilde N \le N$ of non-zero fields that overlap in a multiple junction. In Fig. 6.1 $\tilde N=3$ in the triple junction in the center.

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Fig. 6.1

Scheme of a triple junction between three phase fields. Within the bulk regions, only the information of the phase-field index has to be stored to identify the phase. Within the diffuse interface, two phase indices and one value of the phase field have to be stored. In junctions, we need the information of all phase-field indices, and all phase fields need to be stored. Of course, all of this information evolves dynamically during the calculation when the interfaces move

Within this lecture, we will only deal with “junctions,” since the rest is known from dual phase fields as treated in the previous lectures. There is nonetheless enough to do...

The first thing to do is to define the free-energy density of a multi-phase-field model:

$$\displaystyle \begin{aligned} f = \sum_{\alpha=1}^{\tilde N} \left\{ \sum_{\beta=\alpha+1}^{\tilde N} \frac{8}{\pi^2} \frac{\sigma_{\alpha\beta}}{\eta} \left[-\eta^2\nabla\phi_{\alpha}\nabla\phi_{\beta} + \pi^2|\phi_\alpha\phi_\beta| \right] +\phi_\alpha f^{\mathrm{bulk}}_\alpha \right\}. {} \end{aligned} $$

(6.1)

We sum over all $\tilde N$ phases/grains present within the junction for both the capillarity term, or interface, and the bulk free energy $f^{\mathrm {bulk}}_\alpha$, weighted by the local fraction of this phase α given by the phase field ϕ_α. This bulk free-energy density will be a function of temperature, concentration, stress, and strain, and it may contain magnetic or other contributions [cf. also Eq. (2.9) in Chap. 2].

The capillarity term is more complicated; it is expanded in pairs of phases, i.e., we have a second sum over all $\tilde N$ phases. Note that the second sum runs only from $\beta = \alpha +1 \ldots \tilde N$ so as not to count contributions twice. There is also no diagonal term β = α because this form does not relate to an interface between two phases. For triple junctions between three phase fields (usually a “triple line” in 3D), we find three pairs of phases. For quadruple junctions between four phases, we already have six pairs, and there are ten pairs for $\tilde N =5$ and so on. The junctions are then modeled by the superposition of dual interface contributions with the interface energy of a pair σ_αβ. In general, we have $\tilde N$ combinatorial 2 interfaces for $\tilde N$ phases.

There may be higher-order contributions, i.e., contributions where more than two phase fields and their gradients are considered simultaneously. They should give a special penalty to multiple junctions, as discussed by Miyoshi and Takaki [7]. The present form (6.1) is just a pragmatic setting whereby all parameters are, in principle, well-defined physical entities. The interface width η is not considered a physical entity at the mesoscopic scale and is set to a constant value. The interface density $\left [-\eta ^2\nabla \phi _{\alpha }\nabla \phi _{\beta } + \pi ^2|\phi _\alpha \phi _\beta | \right ]$ is expanded in pairs as a pragmatic choice. One can easily check (and please do so) that this form reduces in the case of a dual interface $\tilde N=2$ to the standard form with a double-obstacle potential (2.35) where the bulk free energy is weighted linearly in ϕ_α.

2 Double Obstacle versus Double Well in Multi Phase Field

You may skip this part on first reading, but it is important! Why do we choose a double-obstacle potential instead of a double-well potential? An important feature of the multi-phase-field theory [12] is the sum constraint in junctions: $\sum _{\alpha =1}^{\tilde N} \phi _\alpha = 1$. For the double-well potential we write the potential (without prefactors) $f^{\mathrm {DW}}_0$:

$$\displaystyle \begin{aligned} f^{\mathrm{DW}}_0= \sum_{\alpha=1\ldots\tilde N, \beta=\alpha+1\ldots\tilde N} \phi_\alpha^2\phi_\beta^2. {}\end{aligned} $$

(6.2)

The maximum of the potential in the center of the junction, where $\phi ^\alpha = \phi ^\beta = \ldots = \dfrac 1{\tilde N}$ for all α, β, …, is then evaluated:

$$\displaystyle \begin{aligned} f^{\mathrm{DW}}_0= \binom{\tilde N}{2} (\frac 1 {\tilde N})^4 \propto (\frac 1{\tilde N})^2 \;\; \mbox{for} \;\; \tilde N \gg 1. {}\end{aligned} $$

(6.3)

This means that for $\tilde N>3$, the energy of the junction decreases with the order $\tilde N$ and approaches 0 for large $\tilde N$. This must be termed “unphysical,” since junctions between objects lose their penalty, and the system would return to the disordered state. The double-obstacle potential is introduced [12] to remedy this problem:

$$\displaystyle \begin{aligned} f^{\mathrm{DO}}_0 = \sum_{\alpha=1\ldots\tilde N, \beta=\alpha+1\ldots\tilde N} |\phi_\alpha \phi_\beta|. {} \end{aligned} $$

(6.4)

This has the same topology as the double-well potential (see Fig. 2.2) but a maximum power of two. We calculate the maximum potential of the junction $f^{\mathrm {DO}}_m$:

$$\displaystyle \begin{aligned} f^{\mathrm{DO}}_0= \binom{\tilde N}{2} (\frac 1 {\tilde N})^2 \propto 1 \;\; \mbox{for} \;\; \tilde N \gg 1. {} \end{aligned} $$

(6.5)

This means that the energy of the junction increases with the order $\tilde N$ and approaches a constant for large $\tilde N$, as it should. The main drawback of this potential is the non-analytical form with the absolute signs. But this is technical...

3 Multi-Phase-Field Equation

The most important feature of the multi-phase-field theory is the consideration of a sum constraint:

$$\displaystyle \begin{aligned} \sum_\alpha \phi_\alpha = 1. {}\end{aligned} $$

(6.6)

The system must be closed in itself so that “holes” do not arise, and this has important consequences for further model development. The phase fields within a junction cannot be varied independently; $\frac {\delta \phi _\beta }{\delta \phi _\alpha } \ne 0$. Starting with a relaxation ansatz (2.6), we must apply the chain rule:

$$\displaystyle \begin{aligned} \frac{\partial \phi_\alpha}{\partial t}\propto-\left [\frac{\delta }{\delta \phi_\alpha }+\sum_{\beta\ne\alpha}{\frac{\partial \phi_\beta}{\partial \phi_\alpha }\frac{\delta }{\delta \phi_\beta }}\right]F. {}\end{aligned} $$

(6.7)

The variation $\frac {\partial \phi _\beta }{\partial \phi _\alpha }$, however, is generally unknown. Therefore, so-called “interface fields” ψ_αβ are introduced [12]:

$$\displaystyle \begin{aligned} \psi_{\alpha \beta} = \left [ \frac{\delta }{\delta \phi_\alpha } - \frac{\delta }{\delta \phi_\beta } \right ] F. {}\end{aligned} $$

(6.8)

These have the important property of automatically conserving the sum constraint (6.6) (for details see [3, 12]). With the phase-field mobility $M_{\alpha \beta }^\phi$ defined for a pair of phases, we reformulate (6.7):

$$\displaystyle \begin{aligned} \frac{\partial \phi_\alpha}{\partial t}= -\sum_\beta {\frac{M_{\alpha \beta}^\phi}{\tilde N}}\;\; \psi_{\alpha \beta} = -\sum_\beta {\frac{M_{\alpha \beta}^\phi}{\tilde N}} \left [ \frac{\delta }{\delta \phi_\alpha } - \frac{\delta }{\delta \phi_\beta } \right ] F. {} \end{aligned} $$

(6.9)

Now we have consequently decoupled a multi-phase problem into pairwise contributions. We end this lecture with two remarks, or warnings:

• As elegant as it looks, it is tedious to implement in phase-field software: since F is itself expanded in pairs, there is a third loop over phases γ≠α≠β.

• As tedious it is to implement, it is nonetheless crucial if you are considering a real materials problem. Taking any shortcuts to avoid the general implementation will lead to a deadlock regarding the solution of the materials problem.

A kind of benchmark for the equilibrium configuration of junctions is displayed in Fig. 6.2 where a brick-like initial structure relaxes into the equilibrium configuration [3, 4]. Note: This equilibrium configuration does not fit nicely into a rectangular box. Therefore one has to employ some tricks regarding boundary conditions and keeping the center within the box, it has the tendency to drift out, of course to reduce interface energy. Such important details for practical applications, however, are seldom discussed in close detail in the literature.

Fig. 6.2

Force balance of a quadruple junction between four grains with isotropic interface energy. (a) The benchmark test starts with a simple configuration of rectangular grains. The triple lines between three phases are displayed. The inset shows the phase configuration in color coding. (b) By reduction of interface energy, the system relaxes to the final configuration with equal angles of 109^∘ between the triple lines and a symmetric grain structure

4 Exercise

Exercise

Prove that the form (6.8) for the interface field automatically conserves the sum constraint (6.6) by adding a Lagrange multiplier $\lambda \left \{ \sum _\alpha \phi _\alpha - 1\right \}$ to the free energy F, see [12].

Example: Anisotropic Grain Growth

A multi-phase-field 3D grain growth model is formulated in [10] to simulate grain growth with anisotropic interface energy. The interface energy is defined as a function of the inclination angle of the boundary plane. It is found that the resulting grain-growth kinetics are strongly influenced by the anisotropy of the interface energy. This results in a slower growth rate and a distinct morphological evolution, as indicated by the presence of more cubic grains and a more significant number of triple-junction angles between 90^∘ and 180^∘, as illustrated in Fig. 6.3. Additionally, the model closely matches various experimental results for NaCl and MgO polycrystalline minerals, where the distribution of grain-boundary planes peaks at low-index {100}-type boundaries.

Fig. 6.3

Three dimensional simulation of anisotropic grain growth in 3D [10]. A 2D cross-section and the bow up of an insert shows clearly the deviation of the angles of interfaces at junctions from the isotropic angle expected for isotropic grain growth. Also the structure shows a pronounced cubic texture

Further Reading

• A multi-phase-field model without the sum constraint, but with a consistent treatment of the bulk-energy part, is presented in [8, 9].

• Force balance at junctions [3, 14].