1. Macroscopic Dynamics of Ferromagnetic Nematics
Author: Tilen Potisk
Advisor: Doc. Dr. Daniel Svenšek

2. OUTLINE Motivation Theory of Macroscopic Dynamics Experiments
Numerical results
Conclusion

3. MOTIVATION Hypothesized in 1970 (Brochard and de Gennes).
Nematic liquid crystals + magnetic nanoparticles
Magneto optic devices, magnetic field visualization, LCDs, ...
Ferromagnetic liquid crystal
Discovered in 2013 (Mertelj et al)

4. MACROSCOPIC DYNAMICS
Our wish - description of a macroscopic system out of equilibrium.
Only few variables are slow in homogeneous limit:
𝝎 𝒌→𝟎 ≈𝟎
Three types:
Consevation laws (global symmetries) – mass, energy, momentum
Can not be created or destroyed localy (only transported).
Spontaneous symmetry breaking – director
Homogenous rotation of director does not cost energy.
Slowly relaxing variables – magnetization
Slow relaxation

5. MACROSCOPIC DYNAMICS Relevant variables: 𝑛 𝑖 + 𝑌 𝑖 =0 𝑀 𝑖 + 𝑋 𝑖 =0
Mass 𝑚
Momentum 𝑔
Energy 𝜖
Concentration 𝑐
Order parameter 𝑆
Director 𝒏
Magnetization 𝑴
A. Mertelj et al: Magneto-optic and converse magnetoelectric effects in a ferromagnetic liquid crystal
𝑛 𝑖 + 𝑌 𝑖 =0
𝑀 𝑖 + 𝑋 𝑖 =0
A. Mertelj et al: Ferromagnetism in suspensions of magnetic platelets in liquid crystal

6. MACROSCOPIC DYNAMICS 1. step: Describe statics
𝜀=− 𝑀 𝑖 𝐵 𝑖 𝐴 1 𝑛⋅𝑀 𝐴 2 𝑀 𝑖 2 + 𝐾 𝑖𝑗𝑘𝑙 𝛻 𝑗 𝑛 𝑖 𝛻 𝑘 𝑛 𝑙
Coupling: Interaction of nematic with the surface of nanoplatelets
(nanometre sized ferromagnetic platelets).
Elastic term: 𝐾 1 𝛻⋅𝑛 𝐾 2 𝑛⋅ 𝛻×𝑛 𝐾 3 n× 𝛻×𝑛 n× 𝛻×𝑛 2
Coupling
Elastic term
Splay
Twist
Bend

7. MACROSCOPIC DYNAMICS 2. Step: Derive thermodynamic forces
Forces are thermodynamic conjugates (variational derivatives):
ℎ 𝑖 𝑛 = 𝜕𝜀 𝜕 𝑛 𝑖 − 𝛻 𝑗 𝜕𝜀 𝜕 𝛻 𝑗 𝑛 𝑖 ≔ 𝛿𝜀 𝛿 𝑛 𝑖
In equilibrium: ℎ 𝑖 𝑛 =0.
Forces: nematic molecular field, magnetic molecular field, temperature gradient 𝛻𝑇, concentration gradient 𝛻𝑐, density gradient 𝛻𝜌.

8. MACROSCOPIC DYNAMICS 𝑛 𝑖 + 𝑌 𝑖 =0 𝑀 𝑖 + 𝑋 𝑖 =0
𝑛 𝑖 + 𝑌 𝑖 =0
𝑀 𝑖 + 𝑋 𝑖 =0
3. Step: Expand (quasi) currents into forces.
Symmetry of reversible and irreversible (dissipative) currents.
Example: 𝑡→−𝑡, time reversal symmetry, 𝐽= 𝐽 𝐷 + 𝐽 𝑅 .
Entropy production
𝑇 𝑆 ≔ 𝑖 𝐽 𝑖 𝑋 𝑖
In the lowest order 𝑇 𝑆 is a quadratic form in forces (dissipation function):
𝑇 𝑆 = 𝑖,𝑗 𝐿 𝑖𝑗 𝑋 𝑖 𝑋 𝑗 =2𝑅.
Obtain dissipative currents by partial derivation:
𝜕𝑅 𝜕 𝑋 𝑖 = 𝐽 𝑖 𝐷
𝐽 𝑖 = 𝑗 𝐿 𝑖𝑗 𝑋 𝑗

9. MACROSCOPIC DYNAMICS 𝑛 𝑖 + 𝑌 𝑖 =0 𝑀 𝑖 + 𝑋 𝑖 =0
𝑛 𝑖 + 𝑌 𝑖 =0
𝑀 𝑖 + 𝑋 𝑖 =0
Dissipative part of the currents:
𝑌 𝑖 𝐷 = 1 𝛾 1 𝛿 𝑖𝑗 ⊥ ℎ 𝑗 𝑛 + 𝜒 𝑖𝑗 𝐷 ℎ 𝑗 𝑀
𝑋 𝑖 𝐷 = 𝑏 𝑖𝑗 𝐷 ℎ 𝑗 𝑀 + 𝜒 𝑗𝑖 𝐷 ℎ 𝑗 𝑛
𝜒 𝑖𝑗 𝐷 = 𝜒 1 𝐷 𝛿 𝑖𝑘 ⊥ 𝑀 𝑘 𝑛 𝑗 + 𝜒 2 𝐷 𝛿 𝑖𝑗 ⊥ 𝑀 𝑙 𝑛 𝑙
𝑏 𝑖𝑗 𝐷 = 𝑏 ⊥ 𝛿 𝑖𝑗 ⊥ + 𝑏 ∥ 𝑛 𝑖 𝑛 𝑗
𝛿 𝑖𝑗 ⊥ = 𝛿 𝑖𝑗 − 𝑛 𝑖 𝑛 𝑗
Lowest order in allowed symmetries: odd in 𝒏 and 𝑴.
Uniaxial symmetry.
Transverse Kroenecker delta:
Projection on plane perpendicular to 𝑛.

10. MACROSCOPIC DYNAMICS
For reversible currents, require entropy production to be zero
𝑇 𝑆 =0.
Entropy is here a conserved quantity.
𝑌 𝑖 𝑅 = 𝛾 −1 𝑖𝑗 ℎ 𝑗 𝑛 + 𝜒 𝑅 (𝑛× ℎ 𝑀 )
𝑋 𝑖 𝑅 = 𝑏 𝑖𝑗 𝑅 ℎ 𝑗 𝑀 + 𝜒 𝑅 (𝑛× ℎ 𝑛 )
Structure of 𝑏 𝑖𝑗 𝑅 and 𝛾 −1 𝑖𝑗 :
𝜉 𝑖𝑗 𝑅 = 𝜉 1 𝑅 𝜀 𝑖𝑗𝑘 𝑀 𝑘 + 𝜉 2 𝑅 𝜀 𝑖𝑗𝑘 𝑛 𝑘 𝑛 𝑝 𝑀 𝑝 + 𝜉 3 𝑅 ( 𝜀 𝑖𝑝𝑞 𝑀 𝑝 𝑛 𝑞 𝑛 𝑗 − 𝜀 𝑗𝑝𝑞 𝑀 𝑝 𝑛 𝑞 𝑛 𝑖 )
Not present in ordinary nematics or isotropic ferrofluids.
Even in director 𝑛: 0th order, 2nd order
Odd in magnetization 𝑀: 1st order

11. EXPERIMENTS A cell filled with ferromagnetic liquid crystal.
Director parallel to the surface plates.
Sample is quenched to obtain a stable phase.
A. Mertelj et al: Magneto-optic and converse magnetoelectric effects in a ferromagnetic liquid crystal

12. EXPERIMENTS
Magnetic field perpendicular to glass plates was turned on.
Examined with polarizing microscopy.
The sample is put between the polarizer (P) and the analyzer (A).
A. Mertelj et al: Ferromagnetism in suspensions of magnetic platelets in liquid crystal

13. EXPERIMENTS Optical axis defined by director 𝑛.
Extraordinary and ordinary rays.
Normalized phase difference was measured: r B =1− 𝜙(𝐵) 𝜙(0)
𝜙= 𝑘 0 𝑑 0 𝑑 ( 𝑛 𝑒 (𝑧)− 𝑛 0 )dz .
𝑛 𝑒 𝑧 extraordinary refractive index, 𝑛 0 ordinary refractive index, 𝑘 0 wavenumber, 𝑑 cell thickness, 𝜓 – azimuthal angle of magnetization

14. NUMERICAL RESULTS: STATICS
Equilibrium normalized phase difference.
Equilibrium normalized z-component of magnetization: 𝑀 𝑧 𝑀 0 = 0 𝑑 cos 𝜓 , 𝜓 – azimuthal angle of magnetization

15. NUMERICAL RESULTS: DYNAMICS
Director reorientation in magnetic field: 𝑛=( cos 𝜑 sin 𝜃 , sin 𝜑 sin 𝜃 ,cos⁡(𝜃))

16. NUMERICAL RESULTS: DYNAMICS
Response is faster at higher magnetic fields.
A high value of crosscoupling ⇒ overshoot in phase difference
Numerical results
Experiment
A. Mertelj, N. Osterman: experimental results, unpublished
A. Mertelj, N. Osterman: experimental results, unpublished

17. NUMERICAL RESULTS: DYNAMICS
Relaxation time: model function 𝑓 𝑡 = 𝑐 0 [1− 𝑒 − 𝑡 𝜏 𝑠 ]
Experimental observation: 1/𝜏 linear in magnetic field.
Numerical results:
Weak dissipative crosscoupling ⇒ saturation
Strong dissipative crosscoupling ⇒ 1/𝜏 linear in magnetic field
Numerical results
Experiment
A. Mertelj, N. Osterman: experimental results, unpublished

18. CONCLUSION
Ferromagnetic nematic liquid crystals (hypothesized 1970, discovered 2013)
Coupling of 𝒏 and 𝑴 ⟹ nontrivial dynamics.
Relaxation rate (1/𝜏) grows linearly with 𝑩.
Further investigations: linear momentum, fluctuation modes.
Magnetooptic devices, smart fluids