Numerical Analysis of Liquid Crystal Droplets Angelique Morvant - PowerPoint PPT Presentation

Numerical Analysis of Liquid Crystal Droplets Angelique Morvant Joint work with Ethan Seal Mentor: Dr. Shawn Walker July 26, 2017 Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 1 / 19

What is a Liquid Crystal? Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 2 / 19

What is a Liquid Crystal? The liquid crystal state is intermediate between a solid and a liquid Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 2 / 19

What is a Liquid Crystal? The liquid crystal state is intermediate between a solid and a liquid Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 2 / 19

What is a Liquid Crystal? The liquid crystal state is intermediate between a solid and a liquid Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 3 / 19

What is a Liquid Crystal? The liquid crystal state is intermediate between a solid and a liquid Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 4 / 19

Applications LCD displays 3D microlasers Templates for synthesizing nanoparticles Self-assembly of colloidal crystals Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 5 / 19

Modeling Liquid Crystals Models of liquid crystals depend on two parameters: The director n indicates the average orientation of the molecules Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 6 / 19

Modeling Liquid Crystals Models of liquid crystals depend on two parameters: The scalar order parameter s represents the degree of orientational order Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 7 / 19

Modeling Liquid Crystals Models of liquid crystals depend on two parameters: The scalar order parameter s represents the degree of orientational order Note that s and n are both functions of space and time. Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 7 / 19

Modeling Liquid Crystals The parameters s and n can be used to express the energy of the liquid crystal. Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 8 / 19

Modeling Liquid Crystals The parameters s and n can be used to express the energy of the liquid crystal. For example, in the Ericksen model Ω ( κ |∇ s | 2 + s 2 |∇ n | 2 ) dx E [ n , s ] = � Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 8 / 19

Modeling Liquid Crystals The parameters s and n can be used to express the energy of the liquid crystal. For example, in the Ericksen model Ω ( κ |∇ s | 2 + s 2 |∇ n | 2 ) dx E [ n , s ] = � Liquid crystals will exist in the state with minimum energy Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 8 / 19

Modeling Liquid Crystals Goal: Model the equilibrium shapes of liquid crystal droplets suspended in another liquid crystal by minimizing an energy functional. Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 9 / 19

Modeling Liquid Crystals Goal: Model the equilibrium shapes of liquid crystal droplets suspended in another liquid crystal by minimizing an energy functional. Method: 1 Write down the energy functional Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 9 / 19

Modeling Liquid Crystals Goal: Model the equilibrium shapes of liquid crystal droplets suspended in another liquid crystal by minimizing an energy functional. Method: 1 Write down the energy functional 2 Discretize the energy Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 9 / 19

Modeling Liquid Crystals Goal: Model the equilibrium shapes of liquid crystal droplets suspended in another liquid crystal by minimizing an energy functional. Method: 1 Write down the energy functional 2 Discretize the energy 3 Minimize the discrete energy Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 9 / 19

Energy Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 10 / 19

Energy Ericksen Energy Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 10 / 19

Energy Ericksen Energy ◮ Gives the energy of a liquid crystal based on s and n Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 10 / 19

Energy Ericksen Energy ◮ Gives the energy of a liquid crystal based on s and n Allen-Cahn Energy Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 10 / 19

Energy Ericksen Energy ◮ Gives the energy of a liquid crystal based on s and n Allen-Cahn Energy ◮ Energy associated with the mixing of the two liquid crystals Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 10 / 19

Energy Ericksen Energy ◮ Gives the energy of a liquid crystal based on s and n Allen-Cahn Energy ◮ Energy associated with the mixing of the two liquid crystals ◮ Based on phase-field function φ . Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 10 / 19

Energy Ericksen Energy ◮ Gives the energy of a liquid crystal based on s and n Allen-Cahn Energy ◮ Energy associated with the mixing of the two liquid crystals ◮ Based on phase-field function φ . Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 10 / 19

Energy Ericksen Energy ◮ Gives the energy of a liquid crystal based on s and n Allen-Cahn Energy ◮ Energy associated with the mixing of the two liquid crystals ◮ Based on phase-field function φ Anisotropic Surface Tension Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 11 / 19

Energy Ericksen Energy ◮ Gives the energy of a liquid crystal based on s and n Allen-Cahn Energy ◮ Energy associated with the mixing of the two liquid crystals ◮ Based on phase-field function φ Anisotropic Surface Tension ◮ Describes how molecules behave at the boundary of the two liquid crystals Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 11 / 19

Energy Ericksen Energy ◮ Gives the energy of a liquid crystal based on s and n Allen-Cahn Energy ◮ Energy associated with the mixing of the two liquid crystals ◮ Based on phase-field function φ Anisotropic Surface Tension ◮ Describes how molecules behave at the boundary of the two liquid crystals Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 11 / 19

Energy Ericksen Energy ◮ Gives the energy of a liquid crystal based on s and n Allen-Cahn Energy ◮ Energy associated with the mixing of the two liquid crystals ◮ Based on phase-field function φ Anisotropic Surface Tension ◮ Describes how molecules behave at the boundary of the two liquid crystals Total Energy E [ φ, s , n ] = E Ericksen + E Allen − Cahn + E anchoring Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 12 / 19

Energy Minimization Total Energy E [ φ, s , n ] = E Ericksen + E Allen − Cahn + E anchoring We have total energy in terms of s , n , and φ . Now we need to minimize this energy. Hard to do this directly... Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 13 / 19

Energy Minimization 1 Discretize in space using finite element method Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 14 / 19

Energy Minimization 1 Discretize in space using finite element method ◮ n = � n i =1 n i η i , s h = � n i =1 s i η i , φ = � n i =1 φ i η i Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 14 / 19

Energy Minimization 1 Discretize in space using finite element method ◮ n = � n i =1 n i η i , s h = � n i =1 s i η i , φ = � n i =1 φ i η i 2 Take variational derivatives with respect to n , s , and φ Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 14 / 19

Energy Minimization 1 Discretize in space using finite element method ◮ n = � n i =1 n i η i , s h = � n i =1 s i η i , φ = � n i =1 φ i η i 2 Take variational derivatives with respect to n , s , and φ 3 Take gradient descent steps with respect to n , then s , then φ Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 14 / 19

Energy Minimization 1 Discretize in space using finite element method ◮ n = � n i =1 n i η i , s h = � n i =1 s i η i , φ = � n i =1 φ i η i 2 Take variational derivatives with respect to n , s , and φ 3 Take gradient descent steps with respect to n , then s , then φ ◮ Make each variable change in time so that energy decreases Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 14 / 19

Energy Minimization 1 Discretize in space using finite element method ◮ n = � n i =1 n i η i , s h = � n i =1 s i η i , φ = � n i =1 φ i η i 2 Take variational derivatives with respect to n , s , and φ 3 Take gradient descent steps with respect to n , then s , then φ ◮ Make each variable change in time so that energy decreases ◮ Discretize resulting equation with respect to time Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 14 / 19