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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: This music video provides a nice summary of our work so far (vimeo link).

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


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