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Introduction to Mathematical Modeling for Liquid Crystals

Liquid crystals (LCs) are an awesome material with interesting physics and applications such as LCDs (read more about LCs on Wikipedia).
Nematic LC molecules look like elongated rods:
Nematic LCs                    Smectic Phases
The orientation of the molecules affects the material's behavior, especially optical properties such as birefringence:
Schlieren Texture                    LCD Display
which make LCs useful for Liquid Crystal Displays (LCDs).

We can model (at the macroscopic level) the orientation of LC molecules by an arrow called the director:
LC Molecule Vector/Arrow                       LC degree of orientation
Actually, we use a vector field (lots of little arrows) to represent the orientation of lots of LC molecules (see the simulation figures below). And we can model the "degree of orientation" (how well aligned the molecules are) by a scalar order parameter "s".

The following papers describe a particular model and numerical method to simulate the equilibrium state of liquid crystals:
  • R. H. Nochetto, S. W. Walker, W. Zhang
    Numerics for Liquid Crystals with Variable Degree of Orientation
    MRS proceedings, Symposium NN – Math. and Comp. Aspects of Mater. Sci., vol. 1753, 2015.
  • R. H. Nochetto, S. W. Walker, W. Zhang
    A Finite Element Method For Nematic Liquid Crystals With Variable Degree Of Orientation
    SIAM Journal on Numerical Analysis, vol 55, no 3, pg 1357-1386, June 2017.
  • R. H. Nochetto, S. W. Walker, W. Zhang
    The Ericksen Model of Liquid Crystals with Colloidal and Electric Effects
    Journal of Computational Physics, vol 352, pg 568 - 601, 2018.
  • A. Morvant, E. Seal, S. Walker
    A Coupled Ericksen/Allen-Cahn Model For Liquid Crystal Droplets
    Computers & Mathematics with Applications, vol 75, no 11, pg 4048 - 4065, 2018.
  • A. Diegel, S. Walker
    A Finite Element Method for a Phase Field Model of Nematic Liquid Crystal Droplets
    Communications in Computational Physics, vol 25, pg 155-188, 2019.
  • S. Walker
    A Finite Element Method for the Generalized Ericksen Model of Nematic Liquid Crystals,
    in review.

  • This music video provides a summary of our earlier work on the Ericksen model (vimeo link).

    Music "Undertow" by Duff Paulsen used with permission.
    See below for more simulations and pictures.

    Simulations of the Ericksen Model

    Here are simulations showing the evolution of a liquid crystal director field toward equilibrium using our provably robust gradient flow scheme.

    Simulations of the Uniaxially Constrained Landau-de Gennes Model

    We have extended our method for the Ericksen model to simulate the Landau-de Gennes Model with a uniaxial constraint imposed on the order parameter Q.