Dissolution with a grand-potential formulation

Sharp interface model of dissolution

Stefan problems

\[\begin{split}\frac{\partial c}{\partial t}&=\boldsymbol{\nabla}\cdot(D\boldsymbol{\nabla}\mu)&\qquad\text{on }\Gamma_{l}(t)\,\\ (c-c_{s})v_{n}&=\left.-D\boldsymbol{\nabla}c\cdot\boldsymbol{n}\right|_{l}+\cancel{\left.D_{s}\boldsymbol{\nabla}c\cdot\boldsymbol{n}\right|_{s}}&\qquad\text{on }\Gamma_{ls}(t)\\ \overline{\mu}_{I}-\overline{\mu}^{eq}&={\color{red}-d_{0}\kappa}-\beta v_{n}&\qquad\text{on }\Gamma_{ls}(t)\end{split}\]

where \(c\) and \(c_{s}\) are liquid and solid composition, \(v_{n}=\boldsymbol{v}_{s}\cdot\boldsymbol{n}\) is the normal velocity of interface, \(\overline{\mu}\) is the dimensionless chemical potential \(\overline{\mu}^{eq}\) is the equilibirum chemical potential, \(\kappa\) is the curvature, and \(d_{0}\) is the capillary length.

Grand-potential functional energy

The grand-potential energy \(\Omega\) is composed of two contribution: the interface contribution \(\omega_{int}(\psi,\boldsymbol{\nabla}\psi)\) and the bulk phases \(\omega_{bulk}(\psi,\mu)\):

(616)\[\Omega[\psi,\mu]=\int_{V}\left[\omega_{int}(\psi,\boldsymbol{\nabla}\psi)+\omega_{bulk}(\psi,\mu)\right]dV\]

The interface grand-potential density is

(617)\[\omega_{int}(\psi,\boldsymbol{\nabla}\psi)=H\psi^{2}(1-\psi)^{2}+\frac{\zeta}{2}\bigl|\boldsymbol{\nabla}\psi\bigr|^{2}\]
(618)\[\omega_{bulk}(\psi,\mu)=p(\psi){\color{red}\omega_{l}(\mu)}+\left[1-p(\psi)\right]{\color{red}\omega_{s}(\mu)}\]
(619)\[p_{int}(\psi)=\psi^{2}(3-2\psi)\]

and its derivative w.r.t. \(\psi\) is

\[p^{\prime}(\psi)=6\psi(1-\psi)\]

With that convention, if \(\psi=0\) then \(\omega_{bulk}(\mu)=\omega_{s}(\mu)\) and if \(\psi=1\) then \(\omega_{bulk}(\mu)=\omega_{l}(\mu)\).

We work with the dimensionless composition \(c(\psi,\,\mu)\) describing the local fraction of one chemical species and varying between zero and one. It is related to the concentration \(C(\psi,\,\mu)\) (physical dimension \([\text{mol}].[\text{L}]^{-3}\)) by \(c(\psi,\,\mu)=V_{m}C(\psi,\,\mu)\) where \(V_{m}\) is the molar volume of (\([\text{L}]^{3}.[\text{mol}]^{-1}\)). For both chemical species, the molar volume is assumed to be constant and identical. In the rest of this paper \(V_{m}\) will appear in the equations for reasons of physical dimension, but it will be considered equal to \(V_{m}=1\) for all numerical simulations.

The concentration \(C\) is now a function of \(\psi\) and \(\mu\). It is related to the grand-potential by [1] \(C(\psi,\,\mu)=-\delta\Omega/\delta\mu=-\partial\omega_{bulk}(\psi,\,\mu)/\partial\mu\). The application of that relationship with \(\omega_{bulk}(\psi,\,\mu)\) defined by Eq. (618) yields:

(620)\[C(\psi,\,\mu)=p(\psi)\left[-\frac{\partial\omega_{l}(\mu)}{\partial\mu}\right]+\left[1-p(\psi)\right]\left[-\frac{\partial\omega_{s}(\mu)}{\partial\mu}\right]\]

The concentration \(C(\psi,\mu)\) is defined by an interpolation of derivatives of \(\omega_{s}(\mu)\) and \(\omega_{l}(\mu)\) w.r.t. \(\mu\). Each derivative defines the concentration of bulk phase \(C_{s}(\mu)=-\partial\omega_{s}(\mu)/\partial\mu\) and \(C_{l}(\mu)=-\partial\omega_{l}(\mu)/\partial\mu\).

In Eq. ([eq:Grand-potential_density]), the grand-potential densities of each bulk phase \(\omega_{l}(\mu)\) and \(\omega_{s}(\mu)\) are defined by the Legendre transform of free energy densities \(f_{s}(c)\) and \(f_{l}(c)\):

(621)\[\omega_{\psi}(\mu)=f_{\psi}(c)-\mu C\qquad\text{for}\qquad\psi=s,l\]

where \(\mu=\partial f_{\psi}/\partial C\). Finally, the phase-field equations are obtained from the minimization of the grand-potential functional \(\Omega[\psi,\,\mu]\). The most general PDEs write (see [1], Eq. (43) and Eq. (47)):

(622)\[\frac{\partial\psi}{\partial t}=\mathcal{M}_{\psi}\Bigl\{\zeta\boldsymbol{\nabla}^{2}\psi-H\omega_{dw}^{\prime}(\psi)-p^{\prime}(\psi)\left[\omega_{l}(\mu)-\omega_{s}(\mu)\right]\Bigr\}\]
(623)\[\chi(\psi,\,\mu)\frac{\partial\mu}{\partial t}=\boldsymbol{\nabla}\cdot\left[\mathcal{D}(\psi,\,\mu)\chi(\psi,\,\mu)\boldsymbol{\nabla}\mu\right]-p^{\prime}(\psi)\left[\frac{\partial\omega_{s}(\mu)}{\partial\mu}-\frac{\partial\omega_{l}(\mu)}{\partial\mu}\right]\frac{\partial\psi}{\partial t}\]

Eq. (623) is the evolution equation on chemical potential \(\mu(\boldsymbol{x},\,t)\). It is obtained from the conservation equation \(\partial_{t}C(\psi,\,\mu)=-\boldsymbol{\nabla}\cdot\boldsymbol{j}_{diff}\) where the diffusive flux is given by \(\boldsymbol{j}_{diff}=-\mathcal{D}(\psi,\,\mu)\chi(\psi,\,\mu)\boldsymbol{\nabla}\mu\). The time derivative term has been expressed by the chain rule \(\partial C(\psi,\,\mu)/\partial t=(\partial C/\partial\mu)\partial_{t}\mu+(\partial C/\partial\psi)\partial_{t}\psi\). The function \(\chi(\psi,\,\mu)\), called the generalized susceptibility, is defined by the partial derivative of \(C(\psi,\,\mu)\) with respect to \(\mu\). For most general cases, the coefficient \(\mathcal{D}(\psi,\,\mu)\) is the diffusion coefficient which depends on \(\psi\) and \(\mu\). Here we assume that the diffusion coefficients \(D_{s}\) and \(D_{l}\) are only interpolated by \(\psi\), i.e. \(\mathcal{D}(\psi,\,\mu)\equiv\mathcal{D}(\psi)\). Actually, in next section that equation on \(\mu\) will be transformed back to an equation on \(C\) (or \(c\)) for reasons of mass conservation in simulations. Eq. (620) will be used to supply a relationship between \(\mu\) and \(c\).

Quadratic free energy

If we assume two quadratic free energies for each phase \(\psi=l,s\):

(624)\[f_{\psi}(c)=\frac{\varepsilon_{\psi}}{2}(c-m_{\psi})^{2}+f_{\psi}^{min}\]

where we assumed \(\varepsilon_{s}=\varepsilon_{l}\), then we can derive (see [2] for details) for the equilibrium chemical potential:

\[\overline{\mu}^{eq}=\Delta\overline{f}^{min}/\Delta m\]

and equilibrium compositions for liquid and solid phases:

\[\begin{split}c_{l}^{eq}&=m_{l}+\overline{\mu}^{eq}\\ c_{s}^{eq}&=m_{s}+\overline{\mu}^{eq}\end{split}\]

Equivalent phase-field model

References

Section author: Alain Cartalade