On a nonlocal phase separation model

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On a nonlocal phase separation model

Collaborator: H. Gajewski , K. Zacharias

Description: We consider a binary alloy with components $\;A\;$ and $\;B\;$ occupying a spatial domain $\;\Omega\;$ . We denote by $\;u\;$ and $\;1-u\;$ the (scaled) local concentrations of $\;A\;$ and $\;B\;$ , respectively. Let $\;(0,T)\;$ denote a time interval, $\;\nu\;$ the outer unit normal on the boundary $\;\Gamma = \partial\Omega\;$ , and $\;Q = (0,T) \times \Omega\;$ , $\;\Gamma_T = (0,T) \times \Gamma\;$ .

To describe phase separation in binary systems, one usually uses the Cahn-Hilliard equation. This equation is derived ([1]) from a free energy functional of the form

$\begin{displaymath} F_{CH}(u) = \int_\Omega\Big \{f(u) + \kappa\, u\, (1-u) +\frac{\lambda}{2}\,\Big\vert\nabla u \Big\vert^2 \Big \}dx.\end{displaymath}$ (1)

Here $\;f\;$ is a convex function with the property that $\;f(u) + \kappa u(1-u)\;$ (for sufficiently large $\kappa$ ) forms a so-called double well potential. Minimizing F_CH under the constraint of mass conservation, one identifies the Lagrange multiplier v via the associated Euler-Lagrange equation

$\begin{displaymath} v = f'(u) + \kappa\, (1-2 u) -\lambda \Delta u \end{displaymath}$ (2)

as chemical potential. Now one postulates that $\;- \nabla v\;$ is the driving force for the mass flux $\;{\bf j}\;$ , i.e.

$\begin{eqnarray*} {\bf j} = - \mu \nabla v\end{eqnarray*}$

with a suitable mobility $\;\mu$ . Considering the mass balance, one ends up with the Cahn-Hilliard equation

$\begin{displaymath} \dfrac{\partial u}{\partial t} - \nabla \cdot [\,\mu~ \nabl... ... Q,\quad \nu \cdot (\mu \nabla v) = 0 \;\text{on}\; \Gamma_T ,\end{displaymath}$

(3)

where the Neumann boundary condition guarantees mass conservation

$\begin{eqnarray*} \int_\Omega u(t,x) dx = \int_\Omega u(0,x) dx .\end{eqnarray*}$

Inspecting Cahn-Hilliard's arguments ([1]) establishing (1) as the free energy of the binary system, it seems to be reasonable and even more adequate ([7]) to choose an alternative nonlocal expression like

$\begin{displaymath} F(u) = \int_\Omega\Big \{f(u) +u \int_\Omega {\cal K}(\vert x-y\vert)(1-u(y))\, dy \Big \}dx ,\end{displaymath}$

(4)

where the kernel $\;{\cal K}\;$ of the integral term describes nonlocal interaction. By a simple calculation, we find from (4) the corresponding chemical potential $\;v\;$ as the gradient of $\;F\;$ in the form

$\begin{displaymath} v = f'(u) + w,\quad w(x) = \int_\Omega{\cal K}(\vert x-y\vert)(1-2\,u(y))\, dy. \end{displaymath}$

(5)

Replacing (2) by (5), one gets instead of (3) the problem

$\begin{displaymath} \dfrac{\partial u}{\partial t} - \nabla \cdot (\,\mu \,\nab... ...; Q,\quad \nu \cdot (\mu \nabla v) = 0 \;\text{on}\; \Gamma_T.\end{displaymath}$

(6)

In the standard case $f(u) = u\log u + (1-u)\log(1-u)$ , we have $f'(u) = \log \left(\frac{u}{1-u}\right)$ , and hence by (5)

$\begin{eqnarray*} u= {\cal F}(w-v), ~~{\cal F}(s):=1/(1+e^s).\end{eqnarray*}$

The image of the Fermi function ${\cal F}$ is the interval $\;[0,1]\;$ , so that the nonlocal model automatically satisfies the physical requirement $\;0 \le u(x) \le 1\;$ . This property cannot be guaranteed for solutions of the original Cahn-Hilliard equation, since for fourth-order equations no maximum principle is available. Moreover, by physical reasons it is desirable to admit mobilities $\;\mu\;$ depending on $\;u\;$ and $\;\vert\nabla v\vert$ . A natural choice is $\mu = a(\vert\nabla v\vert)/f''(u)$ ([2], [7]) with a function $\;a\;$ such that $\;s \mapsto a(s)s\;$ is monotone. Then problem (6) can be rewritten as

$\begin{displaymath} \dfrac{\partial u}{\partial t} - \nabla \cdot \bigg [a \lef... ...c{\nabla w}{f''(u)} \right)\bigg] = 0 \;\text{on}\; \Gamma_T. \end{displaymath}$

(7)

It seems worth mentioning that diffusion equations with a nonlocal drift term of this form also model transport processes in semiconductor ([4]) and chemotaxis ([5]) theory.

In this project we proved existence and uniqueness of a global solution u to (7). Moreover, we studied the asymptotic behavior as time going to infinity and characterized the asymptotic state by a variational principle ([6]). An extension to non-isothermal situations was also given ([3]).

References:

J.C. CAHN, J.E. HILLIARD, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), pp. 258-267.
C.M. ELLIOT, H. GARCKE, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), pp. 404-423.
H. GAJEWSKI, On a nonlocal model of non-isothermal phase separation, WIAS Preprint no. 671, 2001, to appear in: Adv. Math. Sci. Appl.
H. GAJEWSKI, K. GRÖGER, Semiconductor equations for variable mobilities based on Boltzmann statistics or Fermi-Dirac statistics, Math. Nachr., 140 (1989), pp. 7-36.
H. GAJEWSKI, K. ZACHARIAS, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), pp. 77-114.
$\dito$ , On a nonlocal phase separation model, WIAS Preprint no. 656, 2001, to appear in: J. Math. Anal. Appl.
G. GIACOMIN, J.L. LEBOWITZ, Phase segregation dynamics in particle systems with long range interactions I. Macroscopic limits, J. Statist. Phys., 87 (1997), pp. 37-61.