Phase Field Modeling Lab: Ferroelectric Switching

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Phase Field Modeling Lab Ferroelectric Switching

• R. Edwin García
• redwing_at_purdue.edu

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Please make sure that you have a copy of
spinodal.py in your working directory
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Definition of Cahn-Hilliard Equation

• doubleWellDerivative A ( 1 - 6 phiVar (1
  - phiVar))
• from fipy.terms.implicitDiffusionTerm import
  ImplicitDiffusionTerm
• from fipy.terms.transientTerm import
  TransientTerm
• diffTerm2 ImplicitDiffusionTerm(coeff
  (diffusionCoeff doubleWellDerivative,))
• diffTerm4 ImplicitDiffusionTerm(coeff
  (diffusionCoeff, epsilon))
• eqch TransientTerm() diffTerm2 - diffTerm4

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Goals for the Day

• Program the Allen-Cahn Equation to simulate
  ferroelectric polarization kinetics
• Initialize the polarization field to a specified
  (random) value

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Free Energy Description of Ferroelectric Materials
f
P
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The Allen-Cahn Equation
(for non-conserved quantities)
this is what you want
this is what you have
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Breaking Down the Problem
source/body force term
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Definition of Cahn-Hilliard Equation

• doubleWellDerivative A ( 1 - 6 phiVar (1
  - phiVar))
• from fipy.terms.implicitDiffusionTerm import
  ImplicitDiffusionTerm
• from fipy.terms.transientTerm import
  TransientTerm
• diffTerm2 ImplicitDiffusionTerm(coeff
  (diffusionCoeff doubleWellDerivative,))
• diffTerm4 ImplicitDiffusionTerm(coeff
  (diffusionCoeff, epsilon))
• eqch TransientTerm() diffTerm2 - diffTerm4

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Breaking Down the Problem
source/body force term
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Things to Keep in Mind in Our Modified Program

• Do you have the correct boundary conditions?
• Do you have the correct material parameters?
  Select numbers for kinetic and equilibrium
  properties
• M 1
• k 10-4
• a -b -1 c 0

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Extending spinodal.py

• Find the location of where the diffusion equation
  was defined
• Identify the term(s) you want to modify
• What is the size of the domain? Try setting the
  domain to be 1x1

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Initializing the Polarization Field
At t 0, the polarization field should be random.
It should take values in the range -1ltPlt1
hint use the random.random() function