This Chapter presents a summary of the Kim-Kim-Suzuki Model. (The Notes are still in making.)
This phase field model was first proposed in the paper1 as a binary model for solidification of multicomponent systems. Since then it has been developed and applied to study and model a range of microstructure evolution phenomenon from eutectic solidification to precipitate growth.
The model is free from the limit in the interface thickness in the WBM model, and the unrealistic assumptions made by Losert, and it correctly generates the solute trapping phenomena at high interface velocity. A few relevant points about the model are:
It is an isothermal model.
Defines n+1 components, and phases, solid and liquid.
\(c_{iS}\), \(c_{iL}\) represent mole fraction of \(i\)’th solute in solid and liquid, they are scalar fields.
The Helmholtz free energy of a phase \(f^p = f(\{c_{ip}\})\), is a function of solute concentrations only.
The interface is defined as a mixture of solid and liquid phases.
The free energy at interface follows the mixture rule : \[f^{interface}=h(\phi)f^S + [1-h(\phi)]f^L\]
The concentration at interface also follows the mixture rule : \[c^{interface}=h(\phi)c^S + [1-h(\phi)]c^L\]
And \(c_{iS}\) and \(c_{iL}\) in a given point are restricted by the equal chemical potential condition.
The material properties are assumed to be independent of the composition.
The total free energy of the system:
\[F=\int_V \left( \frac{\epsilon^2}{2}|\nabla \phi|^2 + wg(\phi) + h(\phi)f^S + [1-h(\phi)]f^L \right)dV\]
where \(\epsilon\) : gradient energy coefficient, \(g(\phi)=\phi^2(1-\phi)^2\) : double well potential and \(w\) : potential height.
Imposing the equal chemical potential condition upon the solid and liquid phases at a point of the system has two advantages over the traditional equal composition condition.
The first is the relaxation of the restriction on the interface width in computation.
The second is that the profile of the equilibrium phase-field gradient becomes symmetric, which suppresses the anomalous nonlinear part in the Gibbs–Thomson effect in the thin interface limit.
The phase-field and diffusion equations from the work of Kim1 and Eiken2 are presented here:
\[\frac{1}{M_\phi} \phi_t = \epsilon^2\nabla^2\phi - W \frac{dg(\phi)}{d\phi} - \frac{dh_p(\phi)}{d\phi}\left( f^S - f^L + \sum_{i=1}^n(c_S - c_L)\tilde{\mu_i} \right)\]
where \(\epsilon\) is the gradient energy coefficient, \(M_\phi\) is the mobility constant and \(W\) is the double well potential barrier height.
The diffusion equation is as such:
\[\frac{\partial c_i}{\partial t} = \nabla \cdot \left( h_d(\phi) \sum_{j=1}^n D^S_{ij} \nabla c_{jS} + [1-h_d(\phi)] \sum_{j=1}^n D^L_{ij} \nabla c_{jL} \right)\]
A point to be noted is that, the concentrations of solid (\(c_S\)) and liquid (\(c_L\)) are defined at a certain infinitesimal point which is assumed to be a mixture of solid and liquid phases. Therefore the equality of chemical potentials imposed is also local. The chemical potential need not stay constant throughout the interfacial region. It is only constant across the interface only at a thermodynamic equilibrium state.
At the interfacial region the components also satisfy the equality of chemical potential and the solute concentration is taken to be
\[c = h(\phi)c_{S} + (1-h(\phi))c_{L}\] \[f^S_{c_S}[c_S(x,t)]=f^L_{c_L}[c_L(x,t)]\]
Assuming \(g(\phi)=\phi^2(1-\phi)^2\), \(\phi_0=1\) (solid) at \(x=-\inf\) and \(\phi_0=0\) (liquid) at \(x=+\inf\), the equilibrium profile solved for 1D is given as:
\[\phi_0=\frac{1}{2}\left( 1 - \tanh\frac{\sqrt{w}}{\sqrt{2\epsilon}} x \right)\]
From \(\phi_0\) we get the the interface energy \(\sigma\) and interface thickness \(2\lambda\) to be \[\sigma=\frac{\epsilon\sqrt{w}}{3\sqrt{2}} \; \; \; \; \; \; 2\lambda=\alpha\frac{\sqrt{2}\epsilon}{\sqrt{w}}\]
where \(\alpha\) is a constant depending on the definition of interface thickness. \(\alpha=2.2\) when \(\phi_0\in(0.1,0.9)\).
When we define a finite interface width, several anomalous interface effects appear. This was shown my Almgren3. The effects can be negated by using interpolation functions with specific symmetry but it appears that all of them cannot be suppressed simultaneously.
For dilute binary alloys with \(D_S<<D_L\), Karma solved this problem by introducing an anti-trapping term in the diffusion equation45. Kim6 extend the anti-trapping method to the cases of the arbitrary multi-component alloys. The assumption \(D_S<<D_L\) is, however, was maintained because it allowed the concentration (or chemical potential) profile at steady state to be determined unambiguously.
The anti-trapping term is introduced to the diffusion equation as:
\[\frac{\partial c_i}{\partial t} = \nabla .[1-h_d(\phi)]\sum_{j=1}^n D_{ij}^L\nabla c_{jL} + \nabla \cdot \alpha_i\frac{\partial \phi}{\partial t}\frac{\nabla \phi}{|\nabla \phi|} \label{eq:compevol}\]
where \(\alpha_i\) is a function of \(c_{iS}\) and \(c_{iL}\), and \(c_i\) is given by \[c_i = h_r(\phi)c_{iS} + (1-h_r(\phi))c_{iL}\]
The three equations, phase evolution, diffusion equation and mass conservation, have three different interpolation functions marked by \(p,d,r\). Even though a single function \(h(\phi)\) must be adopted in the rigorous thermodynamic derivation, it is not the case in mapping the diffuse interface model onto the classical sharp interface model45.
However, a specific symmetry in their functional forms must be imposed in order to suppress the anomalous interface effects, such as interface diffusion and interface stretching. This symmetry condition is equivalent to the requirement that the positions of the effective sharp interfaces for the driving force action (\(h_p\)), diffusivity change (\(h_d\)) and solute partitioning (\(h_r\)) must be in accordance with that of the effective Gibbs–Thomson interface which is the symmetry axis position of the potential \(g(\phi)\). For the potential \(g(\phi)=\phi^2(1-\phi)^2\) having a symmetry axis at \(x=1/2\), the simple interpolation functions such as \(\phi\) satisfy the symmetry condition. Even after the choice of functions, there remains an anomalous interface effect: the chemical potential jump at the effective sharp interface.
Kim6 assumed that the interface width is sufficiently smaller than the diffusion boundary layer width in liquid, that is, the thin interface condition. Note that \(c_{iS}\), \(c_{iL}\) and \(\tilde{\mu_i}\) are constrained by equality of chemical potential. If one of them is known at a given point, the other two at that point are fixed by the condition. For an interface with a finite width, there exist a finite difference between \(c_{iS}^+\) and \(c_{iL}^-\). This makes a corresponding difference in chemical potential, which has been called the chemical potential jump4.5
For multicomponent systems with arbitrary thermodynamic properties, as for dilute binary alloys45 the chemical potential jump can be suppressed by a suitable choice of the interpolation functions and \(\alpha(c_{iS}\), \(c_{iL})\), that is, by balancing the anomalous solute trapping arising from the diffusion through the thick interface.
The procedure for balancing the solute trapping with the anti-trapping current is straightforward.
Find the composition profile \(c_{iL}(x)\) by solving the steady-state diffusion equation.
Extract the straight part from the profile \(c_{iL}(x)\) and then get \(c_{iS}^+\) and \(c_{iL}^-\).
Put \(c_{iS}^+ = c_{iL}^-\) to determine the interpolation functions and \(\alpha(c_{iS}\), \(c_{iL})\) for the condition of vanishing chemical potential jump.
The interpolation functions are found to be \(h_r(\phi)=h_d(\phi)=\phi\) and for a vanishing chemical potential jump the anti-trapping function is given as
\[\alpha_i = \frac{\sqrt{2\omega}}{\epsilon}(c_{iL}-c_{iS})\]
The parameters \(\epsilon\) and \(\omega\) can be found from their relationships with the interface width \(2\xi\) and the interface energy \(\sigma\) in the equilibrium state:
\[2\xi=\frac{\epsilon}{\sqrt{2\varpi}}\int_{\phi_{a}}^{\phi_{b}}\frac{d\phi_{0}}{\phi_{0}(1-\phi_{0})}=\frac{\epsilon}{\sqrt{2\varpi}}ln\frac{\phi_{b}(1-\phi_{a})}{\phi_{a}(1-\phi_{b})}\]
\[\sigma=\epsilon^{2}\int_{-\infty}^{\infty}(\frac{d\phi_{0}}{dx})^{2}dx=\frac{\epsilon\sqrt{\varpi}}{3\sqrt{2}}\]
where \(2\xi\) is defined as the width over which \(\phi\) changes from \(\phi_a\) to \(\phi_b\).
The phase-field mobility \(M_\phi\) has a relationship with the interface mobility \(m\) defined by the ratio between the driving force and the interface velocity. The procedure to find \(M_\phi\) is as such:
Find the profile \(c_{iL}(x)\) under the condition \(c_{iS}^+ = c_{iL}^-\) vanishing chemical potential jump.
Then \(c_{iS}(x)\) and \(\tilde{\mu_i}\) follow from equality of chemical potential constraint.
Insert the composition and chemical potential profiles into the driving force term of the phase-field equation and extract the new driving force for the effective sharp interface with the straight composition profiles in the interfacial region.
The relationship between the new driving force and the interface velocity is found, which yields a relationship between the physical interface mobility \(m\) and the phase-field mobility \(M_\phi\) at the thin interface limit.
Ultimately it leads to:
\[f^{L,e} - f^{S,e} - \sum_{i=1}^n (c^e_{iL} - c^e_{iS}) \tilde\mu_i^0 = V \left( \frac{1}{M_\phi} \frac{\sqrt w}{3 \sqrt2 \epsilon} - a_2 \frac{\epsilon}{\sqrt{2 \sigma}} \zeta \right)\]
where \(\zeta\) is defined as: \[\zeta=\sum^n_{i=1}(c^e_{iL}-c^e_{iS})\sum^n_{j=1}f^{L,e}_{ij}\sum^n_{k=1}d^L_{jk}(c^e_{kL}-c^e_{kS})\]
This can be written into a compact form
\[\zeta = (\Delta c ] [G^L] [D^L]^{-1} [\Delta c) = (\Delta c] [M^L]^{-1} [\Delta c)\] with the matrix notations: \((\Delta c] = c^e_{iL} -c^e_{iS}\), \([G^L] = f^{L,e}_{ij}\) and \([M^L] = M^L_{ij}\).
In particular, for the infinite interface mobility, we obtain the phase-field mobility
\[M_{0\phi} = \frac{W}{3 \cdot \epsilon_\phi^2 \cdot a_2 \cdot \zeta} \hspace{1cm}\]
For binary alloys, the parameter \(\zeta\) becomes \[\zeta=\frac{(c^e_{1L}-c^e_{1S})f^{L,e}_{11}}{D^L_{11}}\]
The KKS model discussed above is expanded to a multi-component system with the following modifications.
The Phase-Evolution-Equation (Allen-Cahn Style) can be derived from the variational derivative of the total free energy as such:
\[\frac{\partial \phi_p }{\partial t} = -\frac{L}{N} \sum_{q\neq p}^{N}{\left[ \frac{\delta F}{\delta \phi_p} - \frac{\delta F}{\delta \phi_q}\right]}\]
The Diffusion-Equation becomes :
\[\frac{\partial c_i }{\partial t} = \nabla \cdot \sum_{j=1}^{k-1} \left( M_{ij}(\phi) \nabla \mu_{ij} \right)\] where \[M_{ij}(\phi) = \sum_{p=1}^{N} \left[ h_p(\phi)\sum_{j=1}\left[D^p_{ik}\frac{d c_k}{d \mu_j}\right] \right] \text{ : } D_{ij} = M_{ik}\frac{\partial f}{\partial c_k \partial c_j}\]
The constraint imposed on the concentration fields are mass conservation and equality of chemical potential implemented as such:
\[ c_j = \sum_{p=1}^{N} h(\phi_p)c^p_j ; \mu_j = \mu_i : \frac{\partial f^p}{\partial c^p_i} = \frac{\partial f^p}{\partial c^p_j} \]
The multi-well potential is chosen to be:
\[g(\phi) = \sum_{i=1}^{N} q \gamma_i \phi_i^2(1-\phi_i)^2 + \sum_{i=1}^{N} \sum_{j>i}^{N} \theta_{ij}\phi_i^2\phi_j^2 + \sum_{i=1}^{N} \sum_{j>i}^{N} \sum_{k>j}^{N} \theta_{ijk} \phi_i^2\phi_j^2\phi_k^2\] or one can use the multi-obstacle potential.
The Phase-Field interpolation function is
\[h(\phi_p) = \phi^3 (1-15\phi + 6\phi^2)\]
1. Kim, S. G., Kim, W. T. & Suzuki, T. Phase-field model for binary alloys. Physical Review E 60, 7186–7197 (1999).
2. Eiken, J., Böttger, B. & Steinbach, I. Multiphase-field approach for multicomponent alloys with extrapolation scheme for numerical application. Physical Review E 73, 066122 (2006).
3. Almgren, R. F. Second-Order Phase Field Asymptotics for Unequal Conductivities. SIAM Journal on Applied Mathematics 59, 2086–2107 (1999).
4. Karma, A. Phase-field formulation for quantitative modeling of alloy solidification. Physical Review Letters 87, 115701–1–115701–4 (2001).
5. 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).
6. Kim, S. G. A phase-field model with antitrapping current for multicomponent alloys with arbitrary thermodynamic properties. Acta Materialia 55, 4391–4399 (2007).