Concentration

Abstract

In this chapter the coupling to solute concentration in an alloy is reviewed. During transformation the concentration between the phases must be redistributed, causing solute diffusion. First different approached to handle this problem are discussed comparatively. Special emphasis is given to the problem of scale in a mesoscopic phase field model. In detail the currently most advanced model, the so-called finite interface dissipation model is explained in detail. The chapter concludes with the discussion of the effect of multi-component diffusion and its handling in a phase-field model. As an example, solidification under additive manufacturing condition is presented with special emphasis of the microsegregation after solidification in a multi-component alloy.

The last chapter discussed the coupling of the phase field to temperature. You will remember that the phase-field functional has three contributions: the “gradient,” the “potential,” and the “bulk free-energy difference.” The gradient has to fight with the other two contributions: the fight with the potential represents capillarity, and the fight of capillarity with the bulk free-energy difference determines phase transformations and growth or shrinkage of phases. Therefore, all these ingredients have to be considered simultaneously.

In this chapter, we consider the effect of solute on phase transformations. The treatment of solute within the diffuse interface of a phase-field model is one of the most controversial issues in phase-field theory. Here, we will give a condensed statement of the problem and then elaborate on the most recent model to couple phase-field theory and solute diffusion within the mesoscopic picture, the so-called “finite interface dissipation” (FID) model [12, 14].

1 Phase Field and Solute Concentration

The problem with treating the phase field and solute concentration in a consistent theory relates to the physical question of how crystal structure and solute composition correlate during a phase transformation. This is the chicken and egg problem: who comes first? During a phase transformation in an alloy, the crystal structure typically changes along with the solubility of the alloying elements. We can likely assume that both mechanisms go hand in hand, but a generally accepted rule cannot be given here. A special case is “spinodal decomposition,” in which the crystal structure stays the same, but due to a miscibility gap, the solute partitions into two composition sets. The theory for the chemical interface energy that dominates in this case was proposed by Cahn and Hilliard [1]. This is now frequently applied in microscopic phase-field models of phase transformations with structural components. This may work well if: (i) the structural part of the transformation is considered together with the chemical interface energy; and (ii) the interface is treated at the atomistic scale.

In a mesoscopic phase-field model, in which the interface width may reach the micrometer scale, a Cahn–Hilliard approach to solute diffusion will cause severe errors because the chemical interface energy in a Cahn–Hilliard model scales with the interface width [11]. If we consider the interface at the atomistic scale, with a roughly 1-nm width compared to a scaled interface width of 1 μm in a mesoscopic simulation, this error will be a factor of 1000. This situation is depicted in Fig. 5.1: the Gibbs energy of a two-phase system f^chem has a hump in the two-phase region between the equilibrium compositions $c^{\mathrm {e}}_\alpha$ and $c^{\mathrm {e}}_\beta$ if the composition is treated as a continuous variable c between $c^{\mathrm {e}}_\alpha$ and $c^{\mathrm {e}}_\beta$. The integral over this hump in the normal direction through the interface gives the chemical interface energy $\sigma _{\mathrm {chem}}=\int dn \; f^{\mathrm {chem}}\propto \eta$.

Fig. 5.1

Sketch of the Gibbs energy of a two-phase system (left). If we map the concentration between the phase concentrations c_α and c_β onto the normal n through the interface, the hump in the chemical Gibbs energy f^chem transfers to a region in space with increased energy (right). This region scales with the interface width η

There are three options to cope with this problem:

• Use adaptive methods to scale down the interface to the atomistic scale.

• Construct a Gibbs-energy landscape without a hump between the phase concentrations.

• Split the composition c into phase compositions c_α.

The first approach, to use adaptive finite elements with mesh refinement towards the interface, raised some interest in the early days of the application of phase fields to solidification problems (see, e.g., [10]). This works sufficiently well in 2D; however, it is hardly possible to resolve structures in the micrometer range in 3D.

The second approach, as applied by Echebarria et al. to dendritic solidification of a binary model alloy [2], is difficult to generalize for an arbitrary Gibbs-energy landscape, and particularly for multicomponent materials. Nevertheless, it presents a rigorous procedure to get rid of spurious effects of coupling between solute redistribution and phase evolution in the interface.

The third approach is mostly used today for applications in multicomponent materials. First introduced by Tiaden et al. [13], this splits the overall composition c into the phase compositions c_α and c_β, where ϕ indexes the α phase:

$$\displaystyle \begin{aligned} {} c = \phi c_\alpha + (1-\phi)c_\beta. \end{aligned} $$

(5.1)

Hereby, the phase compositions c_α and c_β may deviate from the compositions in equilibrium $c^{\mathrm {e}}_\alpha$ and $c^{\mathrm {e}}_\beta$. The extra degree of freedom that arises from this construction is fixed by relating the phase compositions by a partition coefficient taken from the equilibrium phase diagram, as done in the original work [13]. A generalization with an extrapolation scheme to multicomponent phase diagrams is presented by Eiken et al. [3]. An alternative is to postulate local chemical equilibrium within the interface, as proposed by Kim et al. [5], i.e., equal chemical potential: . This is called the “KKS” model from the names of the authors, and it is quite popular. It has specific problems of a technical nature when applied to a general Gibbs-energy landscape. Therefore, Plapp developed the so-called grand-potential approach [8].

The grand potential is a Legendre transformation of the free energy, in which the chemical potential replaces the composition as a state variable. Now, if we assume equal chemical potential within the interface, there is only one variable, , and we are automatically in the minimum state of the free energy within the interface. Thereby, as a side product, the spurious interface energy is gone. Now one solves a diffusion equation for the chemical potential (instead of the diffusion equation for the composition) together with the phase-field equation. The diffusion equation for the chemical potential is formally identical to the diffusion equation for the temperature [Chap. 4, Eq. (4.5)] with a source term proportional to the rate change of the phase field $\dot \phi$. One difficulty remains: how to invert the chemical potential into the phase compositions, which are, of course, the variables of interest if we are investigating a phase transformation in an alloy. An oft-applied remedy is to approximate the Gibbs energies of the individual phases as parabolic. The chemical potential then becomes linear in concentration, and one can directly read the composition in the bulk phases from the chemical potentials.

One severe drawback of all the approaches described above is that they are restricted to local equilibrium within the interface, or at least to “close to local equilibrium.” If this condition breaks down—e.g., in rapid solidification, where solute trapping appears, or in solid-state phase transformations at lower temperatures (such as bainitic or martensitic transformations)—one has to take a different approach.

2 Finite Interface Dissipation Model

The general idea of the FID model [12, 14] is to treat the phase compositions as separate variables within the interface that are connected by the conservation constraint of composition. Technically, one separates the diffusion fluxes into long-range fluxes within the bulk phases and into short-range redistribution fluxes between the individual phases within the interface. Then one can start from arbitrary initial phase-composition conditions; they shall converge toward local equilibrium within the interface if the kinetics of the process permits. We also assume that, in reality, the composition within the interface is not strictly in local equilibrium.

To build such a model, we start from the following assumptions:

• The phase compositions c_α within the interface are a constraint to the mixture composition c =∑_αϕ_αc_α. For a solid–liquid interface, this is simply c = ϕ_Sc_S + ϕ_Lc_L

• The phase concentrations are treated as independent variables, only subject to the sum constraint above.

In a mesoscopic model, the interface is taken as an effective reference volume without specifying its actual position and orientation. Within this reference volume, we take the two phases as “mixed” and allow them to exchange solute to lower the total Gibbs energy of the reference volume. We define the chemical free-energy density f^chem as a weighted sum of the chemical free energies of the individual phases $f^{\mathrm {chem}}_{\alpha / \beta }$:

$$\displaystyle \begin{aligned} {} f^{\mathrm{chem}} = \phi_\alpha f^{\mathrm{chem}}_\alpha + \phi_\beta f^{\mathrm{chem}}_\beta + \lambda \left ( c - \phi_\alpha c_\alpha + \phi_\beta c_\beta \right ), \end{aligned} $$

(5.2)

where the Lagrange multiplicator λ ensures conservation of the concentration and gives us the possibility to take the phase concentrations c_α fully as independent variables. Then, we define a redistribution flux between the phases from the demand of energy reduction with the permeability P (for details see [12]):

$$\displaystyle \begin{aligned} \phi_\alpha \dot{c}_\alpha = - P \frac{\delta}{\delta c_\alpha} F= - P \frac{\partial}{\partial c_\alpha} f^{\mathrm{chem}} = - P \left [\phi_\alpha\frac{\partial f_\alpha}{\partial c_\alpha} - \phi_\alpha \lambda \right], {} \end{aligned} $$

(5.3)

$$\displaystyle \begin{aligned} \phi_\beta \dot{c}_\beta = - P \frac{\delta}{\delta c_\beta} F = - P \frac{\partial}{\partial c_\beta} f^{\mathrm{chem}} = - P \left [\phi_\beta \frac{\partial f_\beta}{\partial c_\beta} - \phi_\beta \lambda \right]. {} \end{aligned} $$

(5.4)

The permeability is the inverse of the resistivity of the interface against redistribution between the different phases. We can certainly assume that the interface has an influence on the diffusion of solutes, that it hinders diffusion or even makes it easier because of free volume within the interface region. This will depend on the particular kind of interface. For a high permeability, it is easy to exchange solute between the phases. A low permeability will characterize a passivated interface through which no solute can pass.

From the conservation condition $\dot c = 0$, which shall hold within one reference volume without external fluxes (we will add these later), one determines the Lagrange multiplier:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} 0 & =&\displaystyle \dot{\left(\phi_\alpha c_\alpha + \phi_\beta c_\beta \right)} = \dot \phi_\alpha c_\alpha + \dot \phi_\beta c_\beta + \phi_\alpha \dot c_\alpha + \phi_\beta \dot c_\beta \\ & =&\displaystyle \dot \phi_\alpha c_\alpha + \dot \phi_\beta c_\beta + \phi_\alpha \dot c_\alpha \\ & &\displaystyle - P \left [ \phi_\alpha \frac{\partial}{\partial c_\alpha} f^{\mathrm{chem}} + \phi_\beta \frac{\partial f_\beta}{\partial c_\beta} - (\phi_\alpha+\phi_\beta) \lambda \right]. \end{array} \end{aligned} $$

(5.5)

$$\displaystyle \begin{aligned} \lambda = \phi_\alpha \frac{\partial f_\alpha}{\partial c_\alpha} + \phi_\beta \frac{\partial f_\beta}{\partial c_\beta} - \frac{\dot{\phi}_\alpha c_\alpha + \dot{\phi}_\beta c_\beta}{P}. {} \end{aligned} $$

(5.6)

Substituting λ into Eqs. (5.3) and (5.4), we get the evolution equations for each phase inside the reference volume,

$$\displaystyle \begin{aligned} {} \phi_\alpha \dot{c}_\alpha = P \phi_\alpha \phi_\beta \left (\frac{\partial f_\beta}{\partial c_\beta} - \frac{\partial f_\alpha}{\partial c_\alpha} \right) + \phi_\alpha \dot{\phi}_\alpha (c_\beta - c_\alpha), \end{aligned} $$

(5.7)

$$\displaystyle \begin{aligned} {} \phi_\beta \dot{c}_\beta = P \phi_\alpha \phi_\beta \left (\frac{\partial f_\alpha}{\partial c_\alpha} - \frac{\partial f_\beta}{\partial c_\beta} \right ) + \phi_\beta \dot{\phi}_\beta (c_\alpha - c_\beta). \end{aligned} $$

(5.8)

Finally, we add the long-range diffusion in the bulk phases in the standard way. D_α and D_β are the diffusion coefficients, and and are the chemical potentials:

(5.9)

(5.10)

By construction, both equations overlap in the interface, where two mechanisms of solute redistribution are considered: redistribution due to a chemical-potential difference and renormalization if the fraction of phases changes $\dot \phi _\alpha =-\dot \phi _\beta \ne 0$.

Remark

All phase-field implementations that use the splitting of the composition c into phase compositions c_α have to include a redistribution step and a renormalization step. This redistribution procedure is, unfortunately, never discussed in detail in the literature, but it is considered as a technical concern. Partitioning and redistribution is, however, a crucial consideration if you want to implement your own solution.

The diffusion Eqs. (5.9) and (5.10) for the phase compositions α and β converge to the KKS model in the case of a very high permeability P →∞ because the redistribution is then very fast and the chemical potentials and must become equal. In the case of a low permeability, however, the interface concentration and the partitioning in the interface may significantly deviate from local equilibrium. Currently, there is no general model to determine the permeability parameter for a particular interface. The model as presented in the original paper [12] must be considered to be in error, since it has an inverse dependence of the permeability on the interface width, which must be considered as nonphysical in a mesoscopic model. A special model for the permeability in an electrochemical system, derived from experiments, is given in [9].

The model has another intriguing feature regarding the influence of the permeability on the motion of the interface. Going back to the model for the chemical free energy (5.2), we notice first that the Lagrange parameter λ takes the role of an effective chemical potential of the interface. In fact, it is—according to (5.6)—a mixture of the chemical potentials of the individual phases, linear in ϕ. Furthermore, it is also a function of the rate change of the phase field $\dot \phi$! Since all kinetic-model equations—phase-field, temperature, and composition up to now (stress and strain will be added in Chap. 2)—are derived as thermodynamically consistent from the same free-energy functional F, this term will enter all kinetic equations. The phase-field equation [cf. Chap. 2, Eq. (2.36)] becomes:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \dot{\phi}_\alpha & =&\displaystyle - M^\phi \frac {\delta F}{\delta \phi} \\ & =&\displaystyle M^\phi \Bigm\{\sigma^* [\nabla^2 - \frac{\pi^2}{\eta^2}(\frac{1}{2} - \phi)] - \frac{\pi}{\eta}\sqrt{\phi(1-\phi)} \Delta g_{\alpha\beta}^\lambda \Bigm\}, {} \end{array} \end{aligned} $$

(5.11)

(5.12)

Rearrangement of the terms proportional to $\dot {\phi }_\alpha$ finally leads to:

$$\displaystyle \begin{aligned} \dot{\phi}_\alpha = K \Bigm\{ \sigma_{\alpha\beta} [\nabla^2 \phi_\alpha + \frac{\pi^2}{\eta^2}(\phi_\alpha-\frac{1}{2})] - \frac{\pi^2}{8\eta} \Delta g_{\alpha\beta} \Bigm\}, {} \end{aligned} $$

(5.13)

$$\displaystyle \begin{aligned} \begin{array}{rcl} K & =&\displaystyle \frac{P \eta M^\phi}{P \eta + M^\phi (c_\alpha - c_\beta)^2}, {} \end{array} \end{aligned} $$

(5.14)

(5.15)

The phase-field mobility is modified by a mechanism that is related to solute redistribution and diffusion. The interface mobility M^ϕ in the thin-interface limit should be taken as the effective mobility derived in Chap. 4, at least for high permeability P when the model converges to the KKS model. To date, a consistent treatment for the non-equilibrium case (small P) for thin interfaces is still missing.

We see that there are three distinct limits for the new phase-field mobility K:

• P →∞: K → M^ϕ;

• P → 0 ∪ (c_β − c_α)≠0: K → 0;

• P → 0 ∪ (c_β − c_α) = 0: K → M^ϕ.

The first case, with high permeability, leaves the phase-field equation unchanged, and the ID model converges to the KKS model with equal chemical potentials within the interface. The second case represents a passivated interface: no redistribution of solute is allowed, but there is a jump in concentration between the phases. Any transformation is forbidden, and the transformation stops. The most interesting case is the third case: there is no concentration difference between both phases, as in a partitionless transformation like martensitic transformation, and the phase-field mobility is unaffected, i.e., the transformation can proceed with very high speed. Again, no detailed studies of this have been conducted to date.

3 Multicomponent Alloy Transformation

Up to here we were mainly speaking about a binary alloy with one composition variable c. This is important and gives you all the relevant information about “alloy transformation,” including the general problems associated with it as partitioning and getting the phase diagram right. The extension to multicomponent alloys is straightforward (see [14]) from a theoretical point of view. It is more than “involved” from a practical point of view. Technical alloys, steels, nickel based superalloys, aluminum or copper alloys, brass and others, but also minerals and ceramics are multicomponent with 10 or more individual components. Also vacancies have to be considered for completeness. The first challenge which arises is the phase diagram from which we would like to read the equilibrium compositions of two particular phases, see Fig. 1.1: There is no way to display a phase diagram of a 10 component alloy and we need a numerical procedure to do the calculation of equilibria and deviation from equilibrium automatically.

The second challenge is related to redistribution and diffusion during a phase transformation. The individual components i of an n_c-component alloy in phase state α, are not independent but connected by constraint to the sum constraint:

$$\displaystyle \begin{aligned} {} \sum_{i=1}^{n_c} c_\alpha^i=1 \end{aligned} $$

(5.16)

The third challenge, and a challenge in its own right, is the construction of the Gibbs energy landscape of a multicomponent alloy with 10s of possible phases. For the latter we have to rely on the so-called CALPHAD (CALculation of PHAse Diagrams) technique (see e.g. Lukas et al. [7]). Open or commercial databases are available which provide the Gibbs energy of a special phase (stable, metastable or even unstable) as function of temperature, pressure and all available components in a special machine readable format. These you simply insert into the phase-field functional (1.4) to calculate the Gibbs energy difference Δg(ϕ, T, c, 𝜖, …). In the same way one can access the chemical potentials , in the Sect. 5.2. Now they are also multi-component and one has to take the derivative in the directions i [14]. As said, in practice this may become very involved.

The last challenge to be mentioned here is multicomponent diffusion, again a challenge in its own right. We would like to incorporate atomic mobilities from so-called “mobility databases” which accompany thermodynamic databases. We reference here to “Further reading.”

In the literature several approaches to cope with “multicomponent transformations” are established:

• Hard-coding of thermodynamic functions. This will be helpful if you have a low ranked problem, say n < 4, and a suitable analytic description of the Gibbs energy landscape available. Also you will need your own code and sufficient programming skills. And you will have to do the job for each new problem.

• Local parabolic approximation of the Gibbs energy landscape. Local means here in composition space for your given alloy system, temperature and pressure. The approximation you may do with the help of a CALPHAD software, or from a given analytic description. The approach has also the advantage that for this quadratic approximation the chemical potential is linear with the composition!

• Local linearization of the phase diagram [3]. This approach is somehow similar to the previous approach, but quite different in technical aspects. We consider it the most efficient approach for large scale computation, but it may not be suited for larger deviation from local equilibrium at the interface.

• Direct coupling to databases. This is the most general way of incorporating the full CALPHAD scheme into a phase-field calculation. Of course it has the highest computational cost. An example using the OpenPhase software is shown at the end of this chapter (Fig. 5.2).

  Fig. 5.2

  

  Phase-field simulation of directional solidification of a four component Ni-Al-Cr-Ta superalloy under additive manufacturing conditions. Left: Al composition on the surface of the simulation box. Right: Composition profiles for the individual alloy component on a cross section horizontally through the simulation box (position indicated in left figure). Colour bars above each profile display the composition range. The directed dendrites under these conditions show almost no side branches and interdendritic γ′ is precipitated at the end of solidification

4 Exercises

Exercise

Derive the expression for the Lagrange multiplier λ (5.6).

Example—Multicomponent Alloy Solidification

The following example relates to multicomponent alloy solidification in a Ni-based superalloy. A model alloy of the commercial alloy CMSX4 with four components, Ni, Al, Cr and Ta, is constructed such to yield a comparable fraction of secondary γ′ phases at the solvus temperature of this alloy. The solidification conditions relate to additive manufacturing, as in the previous Example in Chap. 4.

Multicomponent Alloy Solidification

Further Reading

• Extrapolation scheme for multicomponent phase equilibria [3].

• Pair-wise multicomponent diffusion approach [4, 6]

• FID model for multicomponent alloy transformation [14] and sublattice ordering [15].