examples.elphf.poisson

A simple 1D example to test the setup of the Poisson equation.

>>> from fipy import CellVariable, Grid1D, DiffusionTerm, Viewer
>>> from fipy.tools import numerix

>>> nx = 200
>>> dx = 0.01
>>> L = nx * dx
>>> mesh = Grid1D(dx = dx, nx = nx)

The dimensionless Poisson equation is

\[\nabla\cdot\left(\epsilon\nabla\phi\right) = -\rho = -\sum_{j=1}^n z_j C_j\]

where \(\phi\) is the electrostatic potential, \(\epsilon\) is the permittivity, \(\rho\) is the charge density, \(C_j\) is the concentration of the \(j^\text{th}\) component, and \(z_j\) is the valence of the \(j^\text{th}\) component.

We will be solving for the electrostatic potential

>>> potential = CellVariable(mesh = mesh, name = 'phi', value = 0.)
>>> permittivity = 1.

We examine a fixed distribution of electrons with \(z_{\text{e}^{-}} = -1\).

>>> class ComponentVariable(CellVariable):
...     def __init__(self, mesh, value = 0., name = '',
...                  standardPotential = 0., barrier = 0.,
...                  diffusivity = None, valence = 0, equation = None):
...         CellVariable.__init__(self, mesh = mesh,
...                               value = value, name = name)
...         self.standardPotential = standardPotential
...         self.barrier = barrier
...         self.diffusivity = diffusivity
...         self.valence = valence
...         self.equation = equation
...
...     def copy(self):
...         return self.__class__(mesh = self.mesh,
...                               value = self.value,
...                               name = self.name,
...                               standardPotential =
...                                   self.standardPotential,
...                               barrier = self.barrier,
...                               diffusivity = self.diffusivity,
...                               valence = self.valence,
...                               equation = self.equation)

Since we’re only interested in a single species, electrons, we could simplify the following, but since we will in general be studying multiple components, we explicitly allow for multiple substitutional species and multiple interstitial species:

>>> interstitials = [
...     ComponentVariable(mesh = mesh, name = 'e-', valence = -1)]
>>> substitutionals = []

Because Poisson’s equation admits an infinite number of potential profiles, we must constrain the solution by fixing the potential at one point:

>>> potential.constrain(0., mesh.facesLeft)

>>> charge = 0.
>>> for Cj in interstitials + substitutionals:
...     charge += Cj * Cj.valence

>>> potential.equation = DiffusionTerm(coeff = permittivity) \
...                      + charge == 0

First, we obtain a uniform charge distribution by setting a uniform concentration of electrons \(C_{\text{e}^{-}} = 1\).

>>> interstitials[0].setValue(1.)

and we solve for the electrostatic potential

>>> potential.equation.solve(var = potential)

This problem has the analytical solution

\[\psi(x) = \frac{x^2}{2} - 2x\]

We verify that the correct equilibrium is attained

>>> x = mesh.cellCenters[0]
>>> analyticalArray = (x**2)/2 - 2*x

>>> print(potential.allclose(analyticalArray, rtol = 2e-5, atol = 2e-5))
1

If we are running the example interactively, we view the result

>>> from fipy import input
>>> if __name__ == '__main__':
...     viewer = Viewer(vars = (charge, potential))
...     viewer.plot()
...     input("Press any key to continue...")

Next, we segregate all of the electrons to right side of the domain

\[\begin{split}C_{\text{e}^{-}} = \begin{cases} 0& \text{for $x \le L/2$,} \\ 1& \text{for $x > L/2$.} \end{cases}\end{split}\]

>>> x = mesh.cellCenters[0]
>>> interstitials[0].setValue(0.)
>>> interstitials[0].setValue(1., where=x > L / 2.)

and again solve for the electrostatic potential

>>> potential.equation.solve(var = potential)

which now has the analytical solution

\[\begin{split}\psi(x) = \begin{cases} -x& \text{for $x \le L/2$,} \\ \frac{(x-1)^2}{2} - x& \text{for $x > L/2$.} \end{cases}\end{split}\]

We verify that the correct equilibrium is attained

>>> analyticalArray = numerix.where(x < L/2, -x, ((x-1)**2)/2 - x)

>>> potential.allclose(analyticalArray, rtol = 2e-5, atol = 2e-5).value
1

and again view the result

>>> from fipy import input
>>> if __name__ == '__main__':
...     viewer.plot()
...     input("Press any key to continue...")

Finally, we segregate all of the electrons to left side of the domain

\[\begin{split}C_{\text{e}^{-}} = \begin{cases} 1& \text{for $x \le L/2$,} \\ 0& \text{for $x > L/2$.} \end{cases}\end{split}\]

>>> interstitials[0].setValue(1.)
>>> interstitials[0].setValue(0., where=x > L / 2.)

and again solve for the electrostatic potential

>>> potential.equation.solve(var = potential)

which has the analytical solution

\[\begin{split}\psi(x) = \begin{cases} \frac{x^2}{2} - x& \text{for $x \le L/2$,} \\ -\frac{1}{2}& \text{for $x > L/2$.} \end{cases}\end{split}\]

We again verify that the correct equilibrium is attained

>>> analyticalArray = numerix.where(x < 1, (x**2)/2 - x, -0.5)

>>> potential.allclose(analyticalArray, rtol = 2e-5, atol = 2e-5).value
1

and again view the result

>>> if __name__ == '__main__':
...     viewer.plot()

Last updated on Jun 26, 2024. Created using Sphinx 7.1.2.