Abstract
In this chapter the coupling of the phase field to a second, external field is discussed for the case of nonisothermal systems. A thermodynamic consistent derivation of the set of equations for the phasefield and temperature field is presented, starting from an entropy functional. In the second part of the chapter the systematic error, which arises in a phasefield model at the mesoscopic scale, is discussed. Finally, the socalled thin interface limit is presented, which is an elegant remedy of this systematic error by an asymptotic matching condition. The example in this section relates to solidification under additive manufacturing conditions.
Keywords
 Phase field
 Thermodynamic consistency
 Entropy production
 Heat diffusion
 Coupled solution of partial differential equations
 GibbsThomson equation
 Diffuse interface
 Asymptotic matching
 Thininterface limit
 Rapid solidification
 Additive manufacturing
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1 Thermodynamically Consistent Derivation of the PhaseField Equation Coupled to Temperature
In the previous chapters, we studied the phasefield 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 solidstate 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 heatconduction 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. [5], 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 freeenergy density f:
The internal energy of a solid is assumed to be linearly dependent on temperature T with constant density ρ, specific heat capacity c_{p}, and latent heat of fusion L. The governing equations for T and ϕ are derived consistently with the principle of entropy production \(\dot S > 0\):
Inserting the energy model (4.2) into (4.4), we obtain:
which is the wellknown heatconduction equation, with thermal conductivity \(\lambda _T = \frac {M_T}{c_p T^2}\). The phasefield Eq. (4.3) is then almost identical to the previously derived equation, e.g., for the doubleobstacle potential (2.36). The only modification is the prefactor \(\frac 1T\), which can be assimilated into the phasefield mobility M^{ϕ}. The thermodynamically consistent derivation of phasefield 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 diffusiontype equations. The interesting feature is the source term in each equation: the release of latent heat in the heatdiffusion 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 100line program (not counting input and output) for a phasefield problem, is given in Appendix A.1 for you to try yourself.
2 ThinInterface Limit
Let us now focus on one specific problem associated with a physically meaningful solution of the aboveoutlined 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 T_{i} is set to capillaritycorrected thermal equilibrium at temperature \(T_{\mathrm {m}} + \frac {\sigma ^* \kappa }{\Delta S_{\mathrm {SL}}} \), in which κ < 0 for a convex dendrite tip and ΔS_{SL} 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:
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 lefthand side of Fig. 4.1, the temperature can be assumed to be uniform. In the sharpinterface 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 [4]):
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 steadystate 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 “diffusioncontrolled limit.”
In phase field, this limit seems difficult, because the phasefield equation for \(\dot \phi \) corresponds to the Gibbs–Thomson equation. The seminal contribution of Karma and Rappel [1, 2] was to realize that the temperaturediffusion equation, or heatconduction equation, can “easily” be integrated into the phasefield equation!
To understand this, we go back to Fig. 4.1: the dotted line indicates the phasefield 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 sharpinterface picture) but is continuous over the interface. Correspondingly, the temperature profile is smooth. However, outside the interface, we demand that both temperatures, the sharpinterface and diffuseinterface temperatures, match. This is called the “asymptotic matching condition,” as applied to solving the temperature profile for a given phasefield solution analytically (in a 1D direction normal to the interface).
We can see directly that, systematically, the temperature of the phasefield model has to be lower (in the case of solidification into an undercooled melt) than the respective sharpinterface 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 phasefield 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 diffusioncontrolled limit of a sharpinterface model, but there is necessarily a spurious undercooling in the case of a diffuseinterface 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 Δg^{numerical}. 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 Δg^{spurious} ≈ Av with a constant A depending only on the materials parameter λ, ΔS_{SL}, 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:
Here, again, we add an “intelligent 0”, 0 = Δg^{numerical} − Δg^{spurious}. 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!
This defines the effective mobility of a diffuseinterface 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 phasefield model, coupled to temperature diffusion, runs with a spurious undercooling and a finite effective mobility, even if the physical mobility M^{ϕ} →∞. It also matches the sharpinterface solution outside the “thin” interface. This is called “thininterface asymptotic.” The driving force Δg^{eff} comprises the physical and the spurious (but necessary) numerical contributions.
3 Exercises
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 phasefield model, the heat conduction equation,
is solved implicitly. Here, Q_{local} 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.
Heat Release During Solidification
Further Reading

The original publication on the thininterface limit is by Alain Karma and WouterJan Rappel [2]. The limit for a phase field coupled to solutal diffusion (see Chap. 5) is presented in [1]. Here, the socalled “antitrapping current” for a model with a strong difference in the diffusion coefficients is also introduced. In alloy solidification, the liquid diffusivity is much larger than the solid diffusivity, and this has important consequences for treating the thininterface limit.
References
A. Karma, Phasefield formulation for quantitative modeling of alloy solidification. Phys. Rev. Lett. 87(11), 115701 (2001)
A. Karma, W.J. Rappel, Quantitative phasefield modelling of dendritic growth in two and three dimensions. Phys. Rev. E 57, 4323–4349 (1998)
I. Steinbach, Phasefield models in materials science. Model. Simul. Mater. Sci. Eng. 17, 073001 (2009)
C. Vuik, Some historical notes on the Stefan problem. Nieuw Archief voor Wiskunde IV 11(2), 157–167 (1993). ISSN: 00289825
S.L.Wang et al., Thermodynamicallyconsistent phasefield models for solidification. Physica D 69, 189–200 (1993)
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Steinbach, I., Salama, H. (2023). Temperature. In: Lectures on Phase Field . Springer, Cham. https://doi.org/10.1007/9783031211713_4
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DOI: https://doi.org/10.1007/9783031211713_4
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