Liquid Crystals
Liquid crystals are materials made of molecules with shapes that allow them to take on intermediate phases between liquid and solid hence the name liquid crystal. [1]
Liquid crystals are the technology behind liquid crystal displays or LCD screens which use electricity to change the orientation of a small layer of liquid crystals in your screen to direct light for making images. Other interesting technology that uses liquid crystals includes: medical biosensors for detecting disease[2], microscopic robots [3], adaptive 3D printing materials[4], smart fabric [5], and more.
Liquid Crystals Mathematical Modeling
Below is a summary of the summary that we provided in this paper on modeling nematic liquid crystals using the Landau-de Gennes
Tensor
The Landau-de Gennes model of liquid crystals is popular because it can describe a large class of material deformations. The model uses a tensor order parameter
Landau-de Gennes Energy
Let
One way to write the elastic energy is
The thermotropic energy is written as
Gradient Flow
System dynamics can be modeled by an
Energy Law
A useful property of the PDE is the following energy law
Designing Numerical Schemes
Several challenges arise when designing numerical schemes for this PDE:
- The problem is very non-linear. Linear schemes are much faster than iterative non-linear schemes, and so it is useful to find a way to approximate the problem with linear terms.
-
The unknowns in
are coupled by several terms. This means that we have to solve a very large linear system which is computationally expensive especially if we want to simulate anything in three dimensions. Therefore, it is useful to decouple the unknowns in the problem. - To recover accurate system dynamics. By this I mean that the numerical scheme should try to model the continuous energy law as closely as possible without adding too much artificial numerical dissipation into the system.
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To have energy stability. This means that the numerical scheme will have a decreasing energy law at the discrete level similar to the continuous problem. It is especially nice to have a scheme that satisfies the discrete energy law for any choice of discrete time step,
, since the numerical dissipation from #3 depends on as well.