FiPy
Version 3.1
examples.cahnHilliard.mesh2DΒΆ
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)
where 
is a conserved order parameter, possibly representing
alloy composition or spin.
The double-well free energy function 
penalizes states with intermediate values of
between 0 and 1. The gradient energy term 
,
on the other hand, penalizes sharp changes of .
These two competing effects result in the segregation
of into domains of 0 and 1, separated by abrupt, but
smooth, transitions. The parameters 
and 
determine the relative
weighting of the two effects and 
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 = 1000
... else:
... nx = ny = 20
>>> mesh = Grid2D(nx=nx, ny=ny, dx=0.25, dy=0.25)
>>> phi = CellVariable(name=r"$\phi$", mesh=mesh)
We start the problem with random fluctuations about 
>>> phi.setValue(GaussianNoiseVariable(mesh=mesh,
... mean=0.5,
... variance=0.01))
FiPy doesn’t plot or output anything unless you tell it to:
>>> if __name__ == "__main__":
... viewer = Viewer(vars=(phi,), datamin=0., datamax=1.)
For FiPy, we need to perform the partial derivative
manually and then put the equation in the canonical
form by decomposing the spatial derivatives
so that each Term is of a single, even order:
![\frac{\partial \phi}{\partial t}
= \nabla\cdot D a^2 \left[ 1 - 6 \phi \left(1 - \phi\right)\right] \nabla \phi- \nabla\cdot D \nabla \epsilon^2 \nabla^2 \phi.](../../../_images/math/b0b8bd746e18d11687942a235cf458940baec658.png)
FiPy would automatically interpolate D * a**2 * (1 - 6 * phi * (1 - phi)) onto the faces, where the diffusive flux is calculated, but we obtain somewhat more accurate results by performing a linear interpolation from phi at cell centers to PHI at face centers. Some problems benefit from non-linear interpolations, such as harmonic or geometric means, and FiPy makes it easy to obtain these, too.
>>> PHI = phi.arithmeticFaceValue
>>> D = a = epsilon = 1.
>>> eq = (TransientTerm()
... == DiffusionTerm(coeff=D * a**2 * (1 - 6 * PHI * (1 - PHI)))
... - DiffusionTerm(coeff=(D, epsilon**2)))
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 = 1000.
... else:
... duration = 1000.
>>> while elapsed < duration:
... dt = min(100, numerix.exp(dexp))
... elapsed += dt
... dexp += 0.01
... eq.solve(phi, dt=dt, solver=LinearLUSolver())
... if __name__ == "__main__":
... viewer.plot()
... elif (max(phi.globalValue) > 0.7) and (min(phi.globalValue) < 0.3) and elapsed > 10.:
... break
>>> print (max(phi.globalValue) > 0.7) and (min(phi.globalValue) < 0.3)
True
