examples.cahnHilliard.mesh2DCoupled¶

Solve the Cahn-Hilliard problem in two dimensions.

The spinodal decomposition phenomenon is a spontaneous separation of an initially homogenous mixture into two distinct regions of different properties (spin-up/spin-down, component A/component B). It is a “barrierless” phase separation process, such that under the right thermodynamic conditions, any fluctuation, no matter how small, will tend to grow. This is in contrast to nucleation, where a fluctuation must exceed some critical magnitude before it will survive and grow. Spinodal decomposition can be described by the “Cahn-Hilliard” equation (also known as “conserved Ginsberg-Landau” or “model B” of Hohenberg & Halperin)

$\frac{\partial \phi}{\partial t} = \nabla \cdot D \nabla \left( \frac{\partial f}{\partial \phi} - \epsilon^2 \nabla^2 \phi \right).$

where $\phi$ is a conserved order parameter, possibly representing alloy composition or spin. The double-well free energy function $f = (a^2/2) \phi^2 (1 - \phi)^2$ penalizes states with intermediate values of $\phi$ between 0 and 1. The gradient energy term $\epsilon^2 \nabla^2\phi$ , on the other hand, penalizes sharp changes of $\phi$ . These two competing effects result in the segregation of $\phi$ into domains of 0 and 1, separated by abrupt, but smooth, transitions. The parameters $a$ and $\epsilon$ determine the relative weighting of the two effects and $D$ is a rate constant.

We can simulate this process in FiPy with a simple script:

>>> from fipy import *

(Note that all of the functionality of NumPy is imported along with FiPy, although much is augmented for FiPy‘s needs.)

>>> if __name__ == "__main__":
...     nx = ny = 20
... else:
...     nx = ny = 10
>>> mesh = Grid2D(nx=nx, ny=ny, dx=0.25, dy=0.25)
>>> phi = CellVariable(name=r"$\phi$", mesh=mesh)
>>> psi = CellVariable(name=r"$\psi$", mesh=mesh)

We start the problem with random fluctuations about $\phi = 1/2$

>>> noise = GaussianNoiseVariable(mesh=mesh,
...                               mean=0.5,
...                               variance=0.01).value

>>> phi[:] = noise

FiPy doesn’t plot or output anything unless you tell it to:

>>> if __name__ == "__main__":
...     viewer = Viewer(vars=(phi, psi)) # , datamin=0., datamax=1.)

We factor the Cahn-Hilliard equation into two 2nd-order PDEs and place them in canonical form for FiPy to solve them as a coupled set of equations.

$\frac{\partial \phi}{\partial t} &= \nabla\cdot D \nabla \psi \\ \psi &= \frac{\partial^2 f}{\partial \phi^2} (\phi - \phi^\text{old}) + \frac{\partial f}{\partial \phi} - \epsilon^2 \nabla^2 \phi$

We need to perform the partial derivatives

$\frac{\partial f}{\partial \phi} &= (a^2/2) 2 \phi (1 - \phi) (1 - 2 \phi) \\ \frac{\partial^2 f}{\partial \phi^2} &= (a^2/2) 2 \left[1 - 6 \phi(1 - \phi)\right]$

manually.

>>> D = a = epsilon = 1.
>>> dfdphi = a**2 * 2 * phi * (1 - phi) * (1 - 2 * phi)
>>> dfdphi_ = a**2 * 2 * (1 - phi) * (1 - 2 * phi)
>>> d2fdphi2 = a**2 * 2 * (1 - 6 * phi * (1 - phi))
>>> eq1 = (TransientTerm(var=phi) == DiffusionTerm(coeff=D, var=psi))
>>> eq2 = (ImplicitSourceTerm(coeff=1., var=psi) 
...        == ImplicitSourceTerm(coeff=-d2fdphi2, var=phi) - d2fdphi2 * phi + dfdphi 
...        - DiffusionTerm(coeff=epsilon**2, var=phi))
>>> eq3 = (ImplicitSourceTerm(coeff=1., var=psi) 
...        == ImplicitSourceTerm(coeff=dfdphi_, var=phi)
...        - DiffusionTerm(coeff=epsilon**2, var=phi))

>>> eq = eq1 & eq2

Because the evolution of a spinodal microstructure slows with time, we use exponentially increasing time steps to keep the simulation “interesting”. The FiPy user always has direct control over the evolution of their problem.

>>> dexp = -5
>>> elapsed = 0.
>>> if __name__ == "__main__":
...     duration = .5e-1
... else:
...     duration = .5e-1

>>> while elapsed < duration:
...     dt = min(100, numerix.exp(dexp))
...     elapsed += dt
...     dexp += 0.01
...     eq.solve(dt=dt)
...     if __name__ == "__main__":
...         viewer.plot()

>>> if __name__ == '__main__':
...     raw_input("Coupled equations. Press <return> to proceed...")

These equations can also be solved in FiPy using a vector equation. The variables $\phi$ and $\psi$ are now stored in a single variable

>>> var = CellVariable(mesh=mesh, elementshape=(2,))
>>> var[0] = noise

>>> if __name__ == "__main__":
...     viewer = Viewer(vars=(var[0], var[1]))

>>> D = a = epsilon = 1.
>>> v0 = var[0]
>>> dfdphi = a**2 * 2 * v0 * (1 - v0) * (1 - 2 * v0)
>>> dfdphi_ = a**2 * 2 * (1 - v0) * (1 - 2 * v0)
>>> d2fdphi2 = a**2 * 2 * (1 - 6 * v0 * (1 - v0))

The source terms have to be shaped correctly for a vector. The implicit source coefficient has to have a shape of (2, 2) while the explicit source has a shape (2,)

>>> source = (- d2fdphi2 * v0 + dfdphi) * (0, 1)
>>> impCoeff = -d2fdphi2 * ((0, 0), 
...                         (1., 0)) + ((0, 0), 
...                                     (0, -1.))

This is the same equation as the previous definition of eq, but now in a vector format.

>>> eq = TransientTerm(((1., 0.), 
...                     (0., 0.))) == DiffusionTerm([((0.,          D), 
...                                                   (-epsilon**2, 0.))]) + ImplicitSourceTerm(impCoeff) + source

>>> dexp = -5
>>> elapsed = 0.

>>> while elapsed < duration:
...     dt = min(100, numerix.exp(dexp))
...     elapsed += dt
...     dexp += 0.01
...     eq.solve(var=var, dt=dt)
...     if __name__ == "__main__":
...         viewer.plot()

>>> print numerix.allclose(var, (phi, psi))
True

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