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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?

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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

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What is a Liquid Crystal?

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

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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

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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

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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

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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

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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

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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

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Modeling Liquid Crystals

Models of liquid crystals depend on two parameters: The scalar order parameter s represents the degree of

  • rientational order

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

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Modeling Liquid Crystals

Models of liquid crystals depend on two parameters: The scalar order parameter s represents the degree of

  • rientational 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

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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

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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 E[n, s] =

  • Ω(κ|∇s|2 + s2|∇n|2)dx

Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 8 / 19

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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 E[n, s] =

  • Ω(κ|∇s|2 + s2|∇n|2)dx

Liquid crystals will exist in the state with minimum energy

Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 8 / 19

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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

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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

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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

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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

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Energy

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

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Energy

Ericksen Energy

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

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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

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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

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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

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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

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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

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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

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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

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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

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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] = EEricksen + EAllen−Cahn + Eanchoring

Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 12 / 19

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Energy Minimization

Total Energy E[φ, s, n] = EEricksen + EAllen−Cahn + Eanchoring 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

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Energy Minimization

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

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Energy Minimization

1 Discretize in space using finite element method ◮ n = n

i=1 niηi, sh = n i=1 siηi, φ = n i=1 φiηi

Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 14 / 19

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Energy Minimization

1 Discretize in space using finite element method ◮ n = n

i=1 niηi, sh = n i=1 siηi, φ = n i=1 φiηi

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

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Energy Minimization

1 Discretize in space using finite element method ◮ n = n

i=1 niηi, sh = n i=1 siηi, φ = n i=1 φiηi

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

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Energy Minimization

1 Discretize in space using finite element method ◮ n = n

i=1 niηi, sh = n i=1 siηi, φ = n i=1 φiηi

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

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Energy Minimization

1 Discretize in space using finite element method ◮ n = n

i=1 niηi, sh = n i=1 siηi, φ = n i=1 φiηi

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

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Energy Minimization

1 Discretize in space using finite element method ◮ n = n

i=1 niηi, sh = n i=1 siηi, φ = n i=1 φiηi

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 ◮ Solve repeatedly until energy levels out Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 14 / 19

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Defect Inside a Droplet with Planar Anchoring

Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 15 / 19

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Defect Inside a Droplet with Homeotropic Anchoring

Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 16 / 19

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Conclusions

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

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Conclusions

Our model successfully simulates the effects of the director field

  • n the shapes and positions of liquid crystal droplets

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

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Conclusions

Our model successfully simulates the effects of the director field

  • n the shapes and positions of liquid crystal droplets

Could be useful in applications where the shapes of the droplets are important

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

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Conclusions

Our model successfully simulates the effects of the director field

  • n the shapes and positions of liquid crystal droplets

Could be useful in applications where the shapes of the droplets are important Limitations

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

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Conclusions

Our model successfully simulates the effects of the director field

  • n the shapes and positions of liquid crystal droplets

Could be useful in applications where the shapes of the droplets are important Limitations

◮ Only models liquid crystal droplets suspended in another liquid crystal Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 17 / 19

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Conclusions

Our model successfully simulates the effects of the director field

  • n the shapes and positions of liquid crystal droplets

Could be useful in applications where the shapes of the droplets are important Limitations

◮ Only models liquid crystal droplets suspended in another liquid crystal ◮ Gradient descent method is slow to converge Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 17 / 19

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Acknowledgments

This material is based upon work supported by the National Science Foundation under awards OCI-1560410 and DMI-1555222 (CAREER grant) with additional support from the Center for Computation & Technology at Louisiana State

  • University. Computer support was provided by HPC@LSU computing.

In addition, we would like to thank our mentor Dr. Walker for his guidance.

Morvant, Seal, Walker Numerical Analysis of Liquid Crystal Droplets 7/26/17 18 / 19

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References

  • L. R. Brenner, S. C. & Scott.

The Mathematical Theory of Finite Element Methods. Springer, New York, NY, 2008.

  • I. Humar, M. & Musevic.

3d microlasers from self-assembled cholesteric liquid-crystal microdroplets. Optics Express, 18(26), 2010.

  • G. Lagerwall, J. P. F. & Scalia.

A new era for liquid crystal research: Applications of liquid crystals in soft matter nano-, bio-, and microtechnology. Current Applied Physics, 12:1387–1412, 2012. Ricardo H. Nochetto, Shawn W. Walker, and Wujun Zhang. A finite element method for nematic liquid crystals with variable degree of

  • rientation.

SIAM Journal on Numerical Analysis, 55(3):1357–1386, 2017.

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