PART V.B. Fundamentals of phase-field theory

Fundamentals of phase-field theory

• Principe de moindre action
  • Action \(\mathscr{S}\) et Lagrangien \(\mathcal{L}\)
  • Principe de stationnarité et dérivée variationnelle
  • Équations d’Euler-Lagrange
  • Application: loi du mouvement de Newton
• Fundamentals of phase-field theory
  • Free energy functional with \(\phi\) and double-wells
  • Euler-Lagrange equation and chemical potential
  • Fundamental solution of Euler-Lagrange equation
  • Alternative forms of double-wells
  • Common tangent for a flat interface
  • Properties of equilibrium solutions
  • Appendices
  • References

Dynamics: time-evolution

• Time-evolution: Cahn-Hilliard and Allen-Cahn models
  • Derivation of Cahn-Hilliard (CH) model for conservative order parameter
  • Allen-Cahn model for non-conservative order parameter
  • Derivation of Conservative Allen-Cahn (CAC) model
  • References

Coupling with incompressible Navier-Stokes

• Derivation of incompressible Navier-Stokes with interface-capturing equation model
  • Mass balance & interface equation
  • Derivation of constitutive laws
  • Model of incompressible Navier-Stokes with interface-capturing equation
  • Appendix: details for deriving constitutive laws
  • References

Contact angle and Marangoni force

• Contact angle and Marangoni force
  • Contact angle
  • Marangoni force
  • References

Two-phase with surfactant

• Two-phase with surfactant
  • Free energy
  • Derivation of 1D analytical solution
  • Derivation of counter term for composition equation
  • Model of incompressible two-phase flows with surfactant