This chapter is a compilation of the fundamental concepts in Phase-Field Theory. (The Notes are still in making).
The phase-field is a thermodynamics-based method mostly used to model phase changes and microstructure evolution in materials. It is a mesoscopic method, meaning intermediate size. It deals with material systems between the nano-scale and a few micrometers. The variables used can be abstract non-conserved quantities representing a phase or anything (for example, a variable \(x\) such that \(x=0\) is a solid, \(x=1\) is a liquid and \(x \in (0,1)\) is an interface between the two), or they can be conserved measurable quantity such as concentration.
The principle variable in such a model, that determines the state of a system is called and order parameter.
The state of a system is captured in a variable, which is most often a scalar field, a continuous function of position and time, and that represents a property of the system such as concentration or phase identity. Such a variable is called an Order parameter, and let’s denote it with \(\phi(r,t)\). The value of \(\phi\) at each position and time is determined from the free energy of the system. If in addition of energy conservation, \(\phi(r,t)\) must itself remain conserved throughout its time evolution, it is called a conserved order parameter, for example the concentration field is conserved by the law of conservation of mass. Parameters like phase identity, grain orientation etc. that do not obey any conservation law are called non-conserved order parameters.
The time evolution of a both kinds of order parameters are governed by two different equations.
Let \(\phi\) be the non-conserved order parameter. The free energy density of a phase can be expressed as a function \(f(\phi)\) which depends only on the state of the system (T and P) at each point. Each value of \(\phi\) represents a phase, let’s say \(\phi=0\) is solid and \(\phi=1\) is liquid, then only at the solid-liquid interface \(\phi\) will have a non-zero gradient. In this method, interfaces between phases are detected by taking the gradient of the order parameter, \(\phi\).
The total energy of the system is a sum of both bulk and interfacial energy. Bulk energy distribution is given by a function \(f(\phi)\) and the interfacial energy is assumed to vary linearly with \(|\nabla\phi|^2\). Higher the gradient of \(\phi\), more discontinuous the interface and higher the interfacial energy. The free energy functional is expressed as:
\[F = \int_V \left[ f(\phi) + \frac{\epsilon}{2}\left|\nabla\phi \right|^2 \right] dv\]
where \(\epsilon\) is the constant that determines the interfacial free energy. The functional represents the volumetric total of the free energy of the system. And what should happen to the total free energy as time goes by, it should decrease. So, at equilibrium, \(F\) should be in its minimum value, and the \(\phi\) that minimizes \(F\) is the \(\phi\) at equilibrium. That is the \(\phi\) we are looking for. Now, how to minimize a functional, equate its derivative to 0, yes. But, taking the derivative of a functional has to do with the calculus of variations and is beyond the scope of the present work.
The differential equation obtained as a result of minimizing the total free energy of the system is called the Allen-Cahn equation. This equation governs the evolution of a non-conserved order parameter. It is expressed as:
\[\frac{\partial \phi(r,t)}{\partial t} = -L_\phi \left[ \frac{\partial f(\phi)}{\partial \phi} - \epsilon \nabla^2 \phi(r,t) \right]\]
here \(L_\phi\) is a constant. The key assumptions for this equation to hold are:
These assumptions are also a must for the evolution of conserved order parameters as well.
A conserved order parameter demands a conservation law to be imposed on top of the condition of total free energy minimization of the system. The conservation of most quantities are governed by the mathematical laws as those of mass conservation, namely the Fick’s laws of diffusion which are expressed as:
\[\frac{\partial \phi}{\partial t} = -\nabla J \hspace{1cm}: J = -M\nabla \mu \hspace{1cm}\]
where J is the mass flux, M is the mobility of the material and \(\mu\) is the chemical potential which is defined as:
\[\mu = \frac{\partial f(\phi)}{\partial \phi} - \epsilon \nabla^2 \phi(r,t)\]
Combining the two conditions, one can derive the differential equation governing the time evolution of a conserved order parameter, which is the Cahn-Hlliard equation. It is expressed as:
\[\frac{\partial\phi(r,t)}{\partial t}=\nabla . \left( M \nabla . \left( \frac{\partial f}{\partial \phi} - \epsilon \nabla^2 \phi \right) \right)\] Where \(M\) is the mobility and \(\epsilon\) determines the interfacial free energy. \(M\) can be taken out of the gradient operator when it is a scalar and then we will get a laplacian operating on the inner brackets.
The simplest of all is the renowned work of Kobayashi1 which is one of the earliest phase-field models for dendritic solidification.
The model includes two order parameters. One is a non-conserved order parameter \(\phi(r,t)\) which takes the value of 1 in solid phase and 0 in liquid phase. The other parameter is the temperature field \(T(r,t)\). Here \(r\) is the spatial position and \(t\) is time.
The free energy functional chosen for this model is the Ginzburg-Landau type free energy including \(m\) as a parameter :
\[F(\phi,m) = \int_V \frac{1}{2} \epsilon^2 |\nabla \phi|^2 + f(\phi,m)dv ;\]
where \(\epsilon\) is a small parameter which determines the thickness of the layer and also controls the mobility of the interface. \(f\) is a double well potential (free energy function) that has local minima at \(\phi=0\) and \(\phi=1\) for each value of \(m\). The specific form of \(f\) taken in this model is :
\[f(\phi,m) = \frac{1}{4}\phi^4 + (\frac{1}{2} - \frac{1}{3}m)\phi^3 + (\frac{1}{4} - \frac{1}{2}m)\phi^2\]
where \(|m|<\frac{1}{2}\). Anisotropy is accounted for by assuming that \(\epsilon\) depends on the direction of the outer normal vector at the interface. Its value is calculated as :
\[\epsilon = \bar{\epsilon}\sigma(\theta)\]
where \(\bar{\epsilon}\) is the mean value and \(\sigma(\theta)\) represents anisotropy in the form :
\[\sigma(\theta) = 1 + \delta \cos{j(\theta - \theta_0)} \; \; \text{ with } \; \; \theta = \arctan{ \left(\frac{\partial \phi / \partial y}{\partial \phi/ \partial x} \right)}\]
where \(\delta\) is the strength of anisotropy and \(j\) is the mode number of anisotropy which takes the value of 4 for cubic lattices and 6 for hexagonal lattices. \(\theta_0\) is the initial offset angle taken as a constant.
The parameter \(m\) is assumed to be dependent on the degree of supercooling an the temperature. The dependency is expressed as :
\[m(T) = \frac{\alpha}{\pi} \arctan{[ \gamma (T_{eq}-T } ]\]
where \(\alpha\) is a positive constant and \(T_{eq}\) is the equilibrium temperature. The Allen-Cahn equation for the evolution of the non-conserved order parameter \(\phi\) takes the respective forms as expressed above.
The temperature field evolution equation is derived from enthalpy conservation and is expressed as :
\[\frac{\partial T}{\partial t} = \nabla^2T + \kappa\frac{\partial \phi}{\partial t}\]
The equation is non-dimentionalized so that the characteristic cooling temperature is 0 and the equilibrium temperature is 1.
The isotropic phase field evolution equation is :
\[\tau\frac{\partial \phi}{\partial t} = \epsilon^2\nabla^2\phi + \phi(1-\phi)(\phi- \frac{1}{2} +m )\]
The phase field \(\phi\) evolution equation for anisotropic solidification is expressed as:
\[\tau\frac{\partial \phi}{\partial t} = \frac{\partial}{\partial y} \left( \epsilon \frac{\partial\epsilon}{\partial\theta} \frac{\partial\phi}{\partial x} \right)- \frac{\partial}{\partial y} \left( \epsilon \frac{\partial\epsilon}{\partial\theta} \frac{\partial\phi}{\partial y} \right) + \nabla . (\epsilon^2\nabla\phi) + \phi(1-\phi)(\phi -\frac{1}{2} +m)\]
This model is henceforth referred to as ‘KOB’.
The next important advance was made by Wheeler, Boettinger and McFadden2 and the model came to be known as WBM. The model describes isothermal phase transitions between ideal binary-alloy liquid and solid phases.
Extending on the KOB model, the free energy of an isothermal solid solution in which the two solutes are assumed to be ideal solutions is derived. It is assumed that the freezing temperature of the solution is between the freezing temperatures of the pure elements (a lens phase diagram). Further, the Helmholtz free energy of the components are assumed to be of the form :
\[f_A(\phi; T) = W_A \int_0^{\phi} p(p-1)\left[p - \frac{1}{2} - \beta_A(T)\right] dp,\] where \(W_A\) is a constant, \(T\) is a isothermal parameter and \(\beta_A(T)\) is a monotonically increasing function of temperature capturing the effects of undercooling. The free energy density of the solution is derived to be :
\[f(\phi, c; T) = cf_B + (1 - c) f_A + \frac{RT}{v_m} \left[ c \ln c + (1 - c) \ln (1 - c) \right]\]
where \(R\) is the universal gas constant and \(v_m\) is the molar volume assumed to be a constant. The free energy functional defined in terms of above equation is :
\[F[\phi, c; T] = \int_{V} \left[ f(\phi, c; T) + \frac{\epsilon^2}{2} |\nabla \phi|^2 \right] dv \label{eq:wbmF}\]
Taking the variational derivative of the free energy functional gives the following governing equations:
\[\frac{\partial \phi}{\partial t} = M_1 \left[ \epsilon^2 \nabla^2 \phi - \left( c \frac{\partial f_B}{\partial \phi} + (1-c) \frac{\partial f_A}{\partial \phi} \right) \right]\]
\[\frac{\partial c}{\partial t} = M_2 \nabla \cdot \left[ c(1 - c) \nabla (f_B - f_A) \right] + D \nabla^2 c\]
where \(D=M_2RT/v_m\) is the diffusivity of the solute \(B\) assumed to be constant and equal in both solid and liquid. And \(M_1\) and \(M_2\) are constants related to mobilities of phase and concentration fields.
The model can reproduce correctly the solute trapping phenomena at a high interface velocity. But the drawback of the model is that the microstructure parameters vary depending on the interface thickness and the presence of anomalous interface effects.
The next change is axiom was when Steinbach3 assumed the interface to be a mixture of solid and liquid phases with different compositions, but constant in their ratio (partition coefficient). The model only worked for dilute alloys.
A different derivation of governing equations came from Losert.4 It was long been realized that the governing equations describing alloy solidification are similar to the ones corresponding to pure material. Using this concept, Losert extended the thin-interface PFM for a pure material to an alloy case. However, the model had two unrealistic assumptions. The first is that the model assumed the liquidus and solidus lines to be parallel. And secondly they assumed a constant solute diffusivity throughout the system like in the WBM model.
Building on the approach of Losert, the first successful and widely adopted model for binary-alloy solidification was presented by Seong Gyoon Kim, Won Tae Kim and Toshio Suzuki.5 They utilized the correspondence between the variables of enthalpy-temperature equations and composition-chemical-potential equations to derive an alloy phase field model from the phase field model of pure elements. An important assumption of the model is the equality of chemical potentials of components at the interface.[^1]
If the PFM interface width is of the order of a real interface, the needed computational mesh size for mesoscale simulation amplifies6. If not, various anomalous interface effects are observed7.
The problem of finite interface width is as such. Generally, a binary PFM model consists of two equations. The phase-field evolution equation and the concentration-field evolution equation (the diffusion equation). There are three parameters in the phase-field equation with definite relationships with interface’s energy, width and mobility. For the interface dynamics of the PFM to correspond to that of a real interface, the relationships between the parameters need to be precisely determined. One way to do so is to set the interface width of the PFM to the value corresponding to that of a real interface. This demands the grid size of the computational mesh to be even smaller, about \(<1nm\). Mesoscale simulations then become almost impossible owing to a needed much larger grid size.
This restriction was overcome in a remarkable work of Karma and Rappel.6 They showed that for the solidification of pure substances which have the same heat capacities and thermal diffusivity in solid and liquid forms, trhe dynamics of an interface with a vanishingly small width (that is, classical sharp interface dynamics) can be described correctly by a PFM with a finite thin interface width, given that a novel relationship between the real interface mobility and phase-field mobility is adopted. The interface width does need to be much smaller than the characteristic length scales of the diffusion field as well as the interface curvature.
The PFMs developed with this concept of finite interface width still suffered from anomalous interface effects. This was shown by Almgren7. The effects include a chemical potential jump at the interface which causes an exaggerated solute-trapping effect. Such effects only become significant with increasing interface width.
All the anomalous interface effects could be suppressed by introducing an antitrapping current term into the diffusion equation8.
A solution was proposed by Karma89. To act against the solute-trapping current driven by the chemical potential gradient, an antitrapping current term was introduced to the diffusion equation. This suppressed the anomalous interface effects.
Following these findings, a model new PFM was proposed10. It is based on the fact that all the anomalous interface effects originate from the finite interface width in the diffusion equation, not in the phase-field equation. The anomalous interface effects can then be suppressed by decoupling the interface width \(2\xi_d\) in the diffusion equation from the width \(2\xi_d\) in the phase-field equation and taking the limit \(2\xi_d \rightarrow 0\). In numerical computations, however, the minimum \(2\xi_d\) value which one can take appears to be slightly above the grid size \(\Delta x\). Therefore the solute-trapping phenomenon persists to a certain extent, as long as \(\Delta x\) in the computation is much thicker than the real interface width.
Next was the derivation of a PFM for multi-component systems with arbitrary thermodynamic properties under the equal chemical potential condition. This was done by Kim11 and is discussed in detail in Chapter KKS.
1. Kobayashi, R. Modeling and numerical simulations of dendritic crystal growth. Physica D: Nonlinear Phenomena 63, 410–423 (1993).
2. Wheeler, A. A., Boettinger, W. J. & McFadden, G. B. Phase-field model for isothermal phase transitions in binary alloys. Physical Review A 45, 7424–7439 (1992).
3. Steinbach, I. & Pezzolla, F. A generalized field method for multiphase transformations using interface fields. Physica D: Nonlinear Phenomena 134, 385–393 (1999).
4. Losert, W. et al. Selection of doublet cellular patterns in directional solidification through spatially periodic perturbations. Phys. Rev. E 58, 7492–7506 (1998).
5. Kim, S. G., Kim, W. T. & Suzuki, T. Phase-field model for binary alloys. Physical Review E 60, 7186–7197 (1999).
6. Karma, A. & Rappel, W. J. Quantitative phase-field modeling of dendritic growth in two and three dimensions. Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics 57, 4323–4349 (1998).
7. Almgren, R. F. Second-Order Phase Field Asymptotics for Unequal Conductivities. SIAM Journal on Applied Mathematics 59, 2086–2107 (1999).
8. Karma, A. Phase-field formulation for quantitative modeling of alloy solidification. Physical Review Letters 87, 115701–1–115701–4 (2001).
9. Echebarria, B., Folch, R., Karma, A. & Plapp, M. Quantitative phase-field model of alloy solidification. Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics 70, 22 (2004).
10. Kim, S. G., Kim, W. T. & Suzuki, T. Phase-field model with a reduced interface diffuseness. Journal of Crystal Growth 263, 620–628 (2004).
11. Kim, S. G. A phase-field model with antitrapping current for multicomponent alloys with arbitrary thermodynamic properties. Acta Materialia 55, 4391–4399 (2007).