## 1 Thermodynamically Consistent Derivation of the Phase-Field Equation Coupled to Temperature

In the previous chapters, we studied the phase-field equation itself; it can be used to propagate planar waves, but also to study the effect of capillarity on the equilibrium structures of crystals. For phase transformations, however, we have to consider the release/consumption of latent heat and solute redistribution in an alloy. In this chapter, we will study the interplay between the growth of a crystal and the temperature field around it. We will do this for a solidification problem, but the situation is not restricted to solidification. During a solid-state transformation, the release of latent heat also has a significant effect. We will do this for a pure substance in which solute redistribution plays no role. However, in general, both fields—temperature and solute (see Chap. 5)—have to be coupled to the evolution of the phases in an alloy. See also “Example—Temperature Evolution During Rapid Solidification” at the end of this chapter.

The Gibbs free energy in the case that we solve the heat-conduction equation is not an appropriate starting point, since the Gibbs energy requires that the temperature be given. Instead, one starts from the entropy functional S. Following Wang et al. , the entropy functional S as the integral of the entropy density s over the domain Ω is defined by the internal energy density e and the free-energy density f:

\displaystyle \begin{aligned} {} S = \int_\Omega s = \int_\Omega \frac {e - f} T, \end{aligned}
(4.1)
\displaystyle \begin{aligned} {} e = \rho c_p T + L (1-\phi). \end{aligned}
(4.2)

The internal energy of a solid is assumed to be linearly dependent on temperature T with constant density ρ, specific heat capacity cp, and latent heat of fusion L. The governing equations for T and ϕ are derived consistently with the principle of entropy production $$\dot S > 0$$:

\displaystyle \begin{aligned} {} \tilde \tau \dot \phi = - \frac{\delta}{\delta \phi} \Bigm( \int_\Omega \frac fT \Bigm)_T = - \frac 1T \left [ \frac{\partial f}{\partial \phi} - \nabla \frac {\partial f}{\partial \nabla \phi} \right ], \end{aligned}
(4.3)
\displaystyle \begin{aligned} \begin{array}{rcl} {} \dot e = - \nabla M_T \nabla \frac {\delta }{\delta e} \Bigm( \int_\Omega \frac eT \Bigm)_{\phi} = - \nabla \left [ M_T \nabla \frac 1 T \right ]. \end{array} \end{aligned}
(4.4)

Inserting the energy model (4.2) into (4.4), we obtain:

\displaystyle \begin{aligned} \begin{array}{rcl} {} \dot e = \rho c_p \dot T - L \dot\phi = \nabla \frac {M_T}{T^2} \nabla T \\ {} \rho c_p \dot T = \nabla \lambda_T \nabla T + L \dot\phi, \end{array} \end{aligned}
(4.5)

which is the well-known heat-conduction equation, with thermal conductivity $$\lambda _T = \frac {M_T}{c_p T^2}$$. The phase-field Eq. (4.3) is then almost identical to the previously derived equation, e.g., for the double-obstacle potential (2.36). The only modification is the prefactor $$\frac 1T$$, which can be assimilated into the phase-field mobility Mϕ. The thermodynamically consistent derivation of phase-field models is of special importance, because it enables the correlation of the model parameters with each other, as well as the establishment of a sound theoretical background in thermodynamics.

The derivation above defines a pair of coupled partial differential equations. Both of these are parabolic diffusion-type equations. The interesting feature is the source term in each equation: the release of latent heat in the heat-diffusion equation, which depends on $$\dot \phi$$, and the driving term for the phase field, which depends on the actual temperature of the interface T. This mutual coupling leads to the morphologically unstable evolution of a dendritic solid–liquid interface. It also poses a challenge for numerical solution schemes of this set of equations. Usually, they are solved explicitly in a staggered scheme, i.e., sequentially: one and then the other. In the literature, more advanced schemes can be found, but these do not prevail in general practice. A simple piece of code in C++, a typical 100-line program (not counting input and output) for a phase-field problem, is given in Appendix A.1 for you to try yourself.

## 2 Thin-Interface Limit

Let us now focus on one specific problem associated with a physically meaningful solution of the above-outlined solidification problem: solidification occurring at relatively high temperatures with high mobility of the solid–liquid interface $$M^\phi _{\mathrm {SL}}$$. Typically, this transformation is treated as “diffusion controlled,” meaning that the interface Ti is set to capillarity-corrected thermal equilibrium at temperature $$T_{\mathrm {m}} + \frac {\sigma ^* \kappa }{\Delta S_{\mathrm {SL}}}$$, in which κ < 0 for a convex dendrite tip and ΔSSL is the entropy of fusion. This condition tells us that the Gibbs–Thomson equation, which we have used to control the speed of the interface, becomes undetermined. As a reminder, the Gibbs–Thomson equation reads:

\displaystyle \begin{aligned} {} v = M^\phi_{\mathrm{SL}}\left(\sigma^* \kappa + \Delta S_{\mathrm{SL}}(T_i-T_{\mathrm{m}})\right). \end{aligned}
(4.6)

For any finite velocity v, the condition $$M^\phi _{\mathrm {SL}}\rightarrow \infty$$ determines only the interface temperature $$T_{\mathrm {i}}=T_{\mathrm {m}} - \frac {\sigma ^* \kappa }{\Delta S_{\mathrm {SL}}}$$, but the velocity cannot be specified. To proceed in this situation, we consider the temperature profile ahead of a growing solidification front, as sketched in Fig. 4.1.

In the growing solid on the left-hand side of Fig. 4.1, the temperature can be assumed to be uniform. In the sharp-interface picture (blue line), there is a kink in the temperature profile, which relates to the release of heat at the solidification front. In this case, it helps that the kink of the temperature at the front can be related to the interface velocity (see the classical Stefan problem ):

\displaystyle \begin{aligned} {} L v = \lambda_{\mathrm{S}} \frac{\partial T}{\partial x}\Big|{}_{\mathrm{S}} + \lambda \frac{\partial T}{\partial x}\Big|{}_{\mathrm{L}}\approx \lambda \frac{\partial T}{\partial x}\Big|{}_{\mathrm{L}}, \end{aligned}
(4.7)

in which |S and |L denote that the temperature gradient is evaluated in the solid or liquid, respectively. Fluxes in the solid can be neglected under steady-state conditions because the solid is uniformly at the melting temperature. Equation (4.7) is a balance equation between heat release due to solidification Lv and heat extraction due to diffusion into the supercooled liquid $$\lambda _{\mathrm {L}} \frac {\partial T}{\partial x}$$. We can see that the velocity is now calculated from heat diffusion instead of being proportional to a thermodynamic driving force (Gibbs–Thomson condition with finite mobility). This is called the “diffusion-controlled limit.”

In phase field, this limit seems difficult, because the phase-field equation for $$\dot \phi$$ corresponds to the Gibbs–Thomson equation. The seminal contribution of Karma and Rappel [1, 2] was to realize that the temperature-diffusion equation, or heat-conduction equation, can “easily” be integrated into the phase-field equation!

To understand this, we go back to Fig. 4.1: the dotted line indicates the phase-field contour, which is diffused over the interface width η; the green line indicates the temperature corresponding to this solution. The release of latent heat is no longer concentrated at the front (producing the kink in the temperature profile in the sharp-interface picture) but is continuous over the interface. Correspondingly, the temperature profile is smooth. However, outside the interface, we demand that both temperatures, the sharp-interface and diffuse-interface temperatures, match. This is called the “asymptotic matching condition,” as applied to solving the temperature profile for a given phase-field solution analytically (in a 1D direction normal to the interface).

We can see directly that, systematically, the temperature of the phase-field model has to be lower (in the case of solidification into an undercooled melt) than the respective sharp-interface temperature. On the one hand, this is not good because it is a systematic error that cannot be avoided, and errors are never good; on the other hand, as long as the temperature outside the interface is correct, we will accept it. This is consistent with the notion that a mesoscopic phase-field model does not claim full physical resolution inside the interface. This systematic error then turns out to be very good from a practical point of view: it helps us to relate the velocity of the interface to undercooling!

There is no physical undercooling of the interface in the diffusion-controlled limit of a sharp-interface model, but there is necessarily a spurious undercooling in the case of a diffuse-interface model. If the right temperature solution as function of the velocity can be calculated analytically, it should coincide with the spurious undercooling of a good numerical solution Δgnumerical. For the analytical solution, please refer to the literature [1,2,3]. Here, we simply state that the spurious undercooling can be (on average over the interface) expanded to lowest order in the front velocity v as Δgspurious ≈ Av with a constant A depending only on the materials parameter λ, ΔSSL, and η. This can be understood easily because the spurious undercooling must vanish if the system reaches equilibrium, i.e., v → 0. With little math, we can reformulate the Gibbs–Thomson equation:

\displaystyle \begin{aligned} \begin{array}{rcl} {} v & =&\displaystyle M^{\phi} \left[ \sigma^*\kappa + \Delta g \right]\\ & =&\displaystyle M^{\phi} \left[ \sigma^*\kappa + \Delta g + 0 \right]\\ & =&\displaystyle M^{\phi} \left[ \sigma^*\kappa + \Delta g + \Delta g^{\mathrm{numerical}}-\Delta g^{\mathrm{spurious}}\right]. \end{array} \end{aligned}
(4.8)

Here, again, we add an “intelligent 0”, 0 =  Δgnumerical − Δgspurious. We assume that our numerics are good and that the numerical solution of the “thin interface temperature” within the interface matches the correct analytical solution (green line in Fig. 4.1). Then the numerical driving force can be read from the numerical solution and the spurious driving force can be handled analytically as a “spurious velocity”. This we do!

\displaystyle \begin{aligned} \begin{array}{rcl} {} v (1+M^{\phi}A) & =&\displaystyle M^{\phi} \left[ \sigma^*\kappa + \Delta g + \Delta g^{\mathrm{numerical}}\right], \\ v & =&\displaystyle \frac{M^{\phi}}{1+M^{\phi}A}\left[ \sigma^*\kappa + \Delta g + \Delta g^{\mathrm{numerical}}\right]. \end{array} \end{aligned}
(4.9)

This defines the effective mobility of a diffuse-interface model, $$M^{\phi }_{\mathrm {eff}}=\frac {M^{\phi }}{1+M^{\phi }A}$$. It is largely determined from diffusion in the dying phase. This means that a phase-field model, coupled to temperature diffusion, runs with a spurious undercooling and a finite effective mobility, even if the physical mobility Mϕ →. It also matches the sharp-interface solution outside the “thin” interface. This is called “thin-interface asymptotic.” The driving force Δgeff comprises the physical and the spurious (but necessary) numerical contributions.

## 3 Exercises

#### Exercise

Repeat (4.8) to (4.9) for yourself!

#### Example—Temperature Evolution During Rapid Solidification

In this example, the system is subjected to the solidification conditions of an additive manufacturing process, i.e., a process with a very high cooling rate and very high thermal gradients. In the coupled phase-field model, the heat conduction equation,

\displaystyle \begin{aligned} {} \rho C_{p} \dot{T} = \nabla \cdot (\lambda \nabla T) + Q_{\mathrm{local}}, \end{aligned}
(4.10)

is solved implicitly. Here, Qlocal handles all kinds of heat sources, including both the release of latent heat and external heat sources. From the simulation results shown in Fig. 4.2, we can see the growth of dendrites in cellular form owing to the very high temperature gradient and cooling rate. The line scan of the temperature plot along the z axis highlights that the dendrite tip is the hottest region during solidification. This is due to the release of latent heat during phase transformation.