Collaborator: H. Gajewski
,
K. Zacharias
Description:
We consider a binary alloy with components
and
occupying a spatial domain
. We denote by
and
the (scaled) local concentrations of
and
, respectively.
Let
denote a time interval,
the outer unit
normal on the boundary
, and
,
.
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}](../2001/img34.gif) |
(1) |
Here
is a convex function with the property that
(for sufficiently large
)
forms a so-called
double well potential. Minimizing FCH 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}](../2001/img38.gif) |
(2) |
as chemical potential.
Now one postulates that
is the driving force for the
mass flux
, i.e.
![\begin{eqnarray*}
{\bf j} = - \mu \nabla v\end{eqnarray*}](../2001/img41.gif)
with a suitable mobility
. 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}](../2001/img43.gif) |
(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*}](../2001/img44.gif)
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}](../2001/img45.gif) |
(4) |
where the kernel
of the integral term describes nonlocal
interaction.
By a simple calculation, we find from (4) the corresponding chemical
potential
as the gradient of
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}](../2001/img49.gif) |
(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}](../2001/img50.gif) |
(6) |
In the standard case
, we have
, and hence by (5)
![\begin{eqnarray*}
u= {\cal F}(w-v), ~~{\cal F}(s):=1/(1+e^s).\end{eqnarray*}](../2001/img53.gif)
The image of the Fermi function
is the interval
, so that the nonlocal model automatically satisfies the
physical requirement
. 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
depending on
and
. A natural choice
is
([2], [7])
with a function
such that
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}](../2001/img62.gif) |
(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.
-
, 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.
LaTeX typesetting by I. Bremer
9/9/2002