# Cholesteric Liquid Crystals

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Cholesteric Liquid Crystals

& Active Emulsions DOTTORANDO: LIVIO NICOLA CARENZA SUPERVISOR: PROF. GONNELLA

Summary 

General Framework

Lattice Boltzmann Method for Liquid Crystals in three-dimensional geometries

Cholesteric Liquid Crystals

Active Emulsions

Active Turbulence

General Framework 

Model & Methods of numerical fluid dynamics: coarse grained approach

General Framework 

Model & Methods of numerical fluid dynamics: coarse grained approach

Generalized Navier-Stokes equation

𝜕𝑡 𝜌 𝑣Ԧ + ∇ ⋅ 𝜌 𝑣Ԧ ⊗ 𝑣Ԧ = −∇𝑝 + ∇ ⋅ 𝜎 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 + 𝜎 𝑐𝑜𝑚𝑝𝑙𝑒𝑥

General Framework 

Model & Methods of numerical fluid dynamics: coarse grained approach

Generalized Navier-Stokes equation

Binary mixtures scalar field 𝜑

𝜕𝑡 𝜌 𝑣Ԧ + ∇ ⋅ 𝜌 𝑣Ԧ ⊗ 𝑣Ԧ = −∇𝑝 + ∇ ⋅ 𝜎 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 + 𝜎 𝑐𝑜𝑚𝑝𝑙𝑒𝑥

Concentration 2

𝜕𝑡 𝜑 + ∇ ⋅ 𝜑 𝑣Ԧ = 𝑀 ∇ 𝜇 ,

𝜇=

𝛿𝐹 𝑏𝑚 𝛿𝜑

General Framework 

Model & Methods of numerical fluid dynamics: coarse grained approach

Generalized Navier-Stokes equation

Binary mixtures

Concentration

scalar field 𝜑 

𝜕𝑡 𝜌 𝑣Ԧ + ∇ ⋅ 𝜌 𝑣Ԧ ⊗ 𝑣Ԧ = −∇𝑝 + ∇ ⋅ 𝜎 𝑣𝑖𝑠𝑐𝑜𝑢𝑠 + 𝜎 𝑐𝑜𝑚𝑝𝑙𝑒𝑥

2

𝜕𝑡 𝜑 + ∇ ⋅ 𝜑 𝑣Ԧ = 𝑀 ∇ 𝜇 ,

𝜇=

Anisotropic contributions: vector/tensor order parameters Ψ

𝜕𝑡 Ψ + 𝑣Ԧ ⋅ ∇Ψ + S = Γ

𝛿𝐹 𝛿Ψ

𝛿𝐹 𝑏𝑚 𝛿𝜑

3d Lattice Boltzmann Method

Boltzmann Equation

Distribution functions 𝑓𝑖 (𝑥Ԧ𝛼 , 𝑡)

Physical observables 𝜌 𝑥Ԧ𝛼 , 𝑡 = σ𝑖 𝑓𝑖 𝑥Ԧ𝛼 , 𝑡

DF dynamics

Discretized time, space and velocity 𝜌𝑣Ԧ = σ𝑖 𝑒Ԧ𝑖 𝑓𝑖 (𝑥Ԧ𝛼 , 𝑡)

Collision (BGK approximation) 1 𝑒𝑞 𝑓𝑖𝑐𝑜𝑙𝑙 𝑥Ԧ𝛼 , 𝑡 = 𝑓𝑖 𝑥Ԧ𝛼 , 𝑡 − 𝑓𝑖 𝑥Ԧ𝛼 , 𝑡 − 𝑓𝑖 (𝑥Ԧ𝛼 , 𝑡) 𝜏

Streaming 𝑓𝑖 𝑥Ԧ𝛼 + 𝑒Ԧ𝑖 Δ𝑡, 𝑡 + Δ𝑡 = 𝑓𝑖𝑐𝑜𝑙𝑙 𝑥Ԧ𝛼 , 𝑡

Equilibrium DF Expansion at II order in 𝑣Ԧ continuum equations

Recover

Parallelization 

Long simulation times

Huge amount of memory

Parallelization 

Long simulation times

Huge amount of memory

Summer school on Parallel Computing – Cineca (BO)

Advanced School on Parallel Computing – Cineca (BO)

Parallelization 

Long simulation times

Huge amount of memory Massage Passing Interface  ~ 40Gb of Memory  ~ 3y of CPU 

Summer school on Parallel Computing – Cineca (BO)

Advanced School on Parallel Computing – Cineca (BO)

Cholesteric Liquid Crystals 

Long chained molecules in solution

Lost traslational order, preserved directional order

Cholesteric Liquid Crystals 

Long chained molecules in solution

Lost traslational order, preserved directional order

Chirality Cholesteric LC

Optical properties

Defects in CLC droplets 

Fundamental for optical properties

Strong dependence on geometry and boundary condition

Relaxing dynamics*

Merging droplets*

Droplets under shear

Switching dynamics under electric field

Active Emulsions 

Active matter: energy injection on small scales 

Bacteria Suspensions

Cytoskeleton extracts

Polar order

Confinment of active behavior 𝑎 𝐹 = න 𝑑𝑉 𝜑 2 𝜑 − 𝜑0 4 𝜑𝑐𝑟 2 𝑘𝜑

2

𝑘𝜑 + ∇𝜑 2

𝛽2 𝑘𝑃

2

Lamellar phase if 𝑎 <

Active stress tensor 𝜎 𝑎𝑐𝑡𝑣𝑒 = −𝜁𝜑𝑃 ⊗ 𝑃

4𝑐

+

𝑐 2 + ∇ 𝜑 2

𝑘𝜑 < 0 , 𝑐 > 0

2

𝛼 𝛼 4 𝑘𝑃 2 + 𝜑𝑃 + 𝑃 + ∇𝑃 2 4 2

2

+ 𝛽𝑃 ⋅ ∇𝜑

Results Activity favours ordering  Transition towards mesoscale turbulence  Wealth of different morphologies 

G.Negro,L.N.Carenza,P.Digregorio,G.Gonnella,A.Lam ura Morphology and flow patterns in highly asymmetric active emulsion, Physica A, Vol. 503, 2018, 464-475

F.Bonelli,L.N.Carenza,G.Gonnella,D.Marenduzzo,E.Orl andini,A.Tiribocchi Lamellar ordering, droplet formation and phase inversion in exotic active emulsions, Accepted with minor revisions Scientific Reports (Nature)

L.N.Carenza,G.Gonnella,A.Lamura,G.Negro,A.Tiriboc chi Lattice Boltzmann Methods and Active Fluids, Submitted to EPJ

Active Turbulence 

Energy injection on small scales

What is new?

Mesoscale turbulence

Active Turbulence 

Energy injection on small scales

What is new? 

KOLMOGOROV TURBULENCE : 

High Reynolds number

Hydrodynamics non-linearities

-5/3 spectrum (universal behavior)

Mesoscale turbulence

Active Turbulence 

Energy injection on small scales

What is new? 

Mesoscale turbulence

KOLMOGOROV TURBULENCE : 

High Reynolds number

Hydrodynamics non-linearities

-5/3 spectrum (universal behavior)

ACTIVE TURBULENCE 

Low Reynolds number

Complex coupling source/sink power terms

No universal behavior

Exams and Schools PhD COURSES 

    

PhD SCHOOLS

How to prepare a technical speech in  Summer school on Parallel Computing – English Cineca (Bologna) Management and knowledge of European research model and promotion  Advanced School on Parallel of research results Computing – Cineca (Bologna) Introduction to parallel Computing and GPU Programming using CUDA  XXX National Seminar of Nuclear and Subnuclear Physics "Francesco C++ Romano" OTRANTO Atom-photon interactions Standard model and beyond Linear stability analysis* Computational fluid dynamic*

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