Alejandro D. Rey Materials Modeling Research Group ...

Materials Modeling Research Group

Flow and Texture Modeling of Liquid Crystalline Materials

Alejandro D. Rey


Department of Chemical Engineering

McGill University

Collaborators at ClemsonBarr von OehsenChristopher Cox

Amod OgaleDan Edie

Rheology of complex fluids: modeling and numericsEcole de Ponts-Peking University Workshop

January 5-9th 2009Paris


Classification

nematic cholesteric

homogeneous/achiralLiquid

periodic/chiral

smectic columnar

Sym

met

ry

brea

king

Crystal230 point groups

Orientational Order

orientational & 1D/2Dpositional order

disorder

Ani

sotr

opic

vis

co-

elas

tic

T C

T C

T C

50 liquidcrystal phases

order


Liquid Crystalline Precursors for Structural Materials


Carbonaceous Mesophases

n

MW=o(500g/mol)polycyclic aromatics

splay twist bend

Elasticity Topological Defects Rheology

1

10

100

1000

0.01 0.1 1 10 100

Shear Rate (1/s)

Visc

osity

(Pa

s) 305 C315 C325 C


Outline1. Introduction: Liquid crystals

2. Modeling Overview

3. Leslie-Ericksen Nematodynamics

4. Landau de Gennes Nematodynamics

5. Applications


Order Parameters: n and Q

( ) 1 3x5 ( / 3) ...4 4 x2

Ψ = + − +π π

u Q : uu I

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

5

10

15

20

25

30

35

Teta

Dis

tribu

tion

Func

tion

T=150

T=296

T=408

T=427 23 1S= cos θ -2 2

n

Orientation n and Alignment S

S=0 S=1/2 S=1

⎛ ⎞⎜ ⎟⎝ ⎠

δQ=S nn-3

Quadrupolarorder

parameter Q

Orientation & alignment Q


Computational Flow-Modeling Paradigms F d: :

dtδ ⎛ ⎞Δ = − ⎜ ⎟δ ⎝ ⎠

QT AQ

s

Fddt

⎛ ⎞⎛ ⎞ −λ⎛ ⎞ ⎜ ⎟⎜ ⎟ = ⊗ δ⎜ ⎟ ⎜ ⎟⎜ ⎟ −λ⎝ ⎠⎜ ⎟ ⎜ ⎟δ⎝ ⎠ ⎝ ⎠

AT aQ β

Q

λ: coupling parameter

Flow-Induced Orientation/Back-Flow Cycle

dd t

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ = ⊗⎜ ⎟ ⎜ ⎟⎜ ⎟ λ⎝ ⎠ ⎝ ⎠⎝ ⎠

AQ

Flow velocity

Flow-induced orientation

Q- velocity

Back-flow

s

F⎛ ⎞− λ⎛ ⎞ ⎛ ⎞ ⎜ ⎟= ⊗ δ⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎜ ⎟⎝ ⎠ δ⎝ ⎠

T

Q

( )sv ⇒ = ∇A v

d / dt F /⇒ δ δQ Q

defects

2o

ef

H EE ;R DR

τ ⎛ ⎞= = → =⎜ ⎟τ ⎝ ⎠


Parametric Space2

H * R= , , , ⎛ ⎞ ⎛ ⎞= =⎜ ⎟⎜ ⎟ξ ⎝ ⎠⎝ ⎠n e S

EE = γτ D = γτR

TUT

microfluidics

nanofluidics

( )eD 1=e(D 0)=

E

R

1/U

phase ordering

( )eD → ∞

FI orientation

FI molecular order

Elastic

Surface anchoring

Elastic

Materials Modeling Research GroupQuijada-Garrido et al., Macromolecules, (2000)

Flow-Alignment

Quijada-Garrido et al., Macromolecules, (2000)

Flow-aligning Non-flow-aligning

3 2

2 3

strainvorticity

α + αλ = =

α − α

0 1< λ <1λ >






5. Applications


Flow-Modeling

nQ


Leslie-Ericksen Nematodynamics

se sv + =Γ Γ 0 sv

svse

se x- ; x h n Γ h n Γ ==

s F d: :dt

δ ⎛ ⎞Δ = − ⎜ ⎟δ ⎝ ⎠nT A

n

s

Fddt

⎛ ⎞ ⎛ ⎞−λ⎛ ⎞⎜ ⎟ ⎜ ⎟= ⊗ δ⎜ ⎟⎜ ⎟ ⎜ ⎟λ −⎝ ⎠ ⎝ ⎠δ⎝ ⎠

AT an β

n


2. Back-flowFour Dynamical Laws

vnv

1. Flow alignment

4. Three Visco-elastic Modes23

11 11

,⎛ ⎞

−⎜ ⎟⎝ ⎠

K αγη

( )22 1, 0−K γ22

33 12

,⎛ ⎞

−⎜ ⎟⎝ ⎠

K αγη

n

1η

n

2η3η

n

η1 > η3 > η2

3. Viscous Anisotropy

3α2α


• Flow alignment

• Shear viscosities

• Back-flow

• Secondary flow

• First normal stress difference

• Shear thinning

Leslie-Ericksen Nematorheology


Rheological Characterization

1

10

100

1000

0.01 0.1 1 10 100

Shear Rate (1/s)

Visc

osity

(Pa

s) 305 C315 C325 C

-150

-100

-50

0

50

100

0.01 0.1 1 10 100

Shear Rate (1/s)

Firs

t Nor

mal

Str

ess

Diff

eren

ce (P

a)

315 C

0.01

0.1

1

10 100 100 0

Er, Ericksen numb er

η

U=3.5R=10000β=−1.3

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

10 100 1000

Er, Ericks en number

U=3.5R=10000β=−1.3

shear viscosity 1st normal stress difference

anisotropy nonlinearity


Landau-de Gennes Nematodynamics

( )s F ˆ: :δΔ = −

δT A Q

Qs

Fˆ c

⎛ ⎞−β⎛ ⎞ ⎛ ⎞ ⎜ ⎟= ⊗ δ⎜ ⎟ ⎜ ⎟ ⎜ ⎟−β⎝ ⎠ ⎜ ⎟⎝ ⎠ δ⎝ ⎠

AaTQ

Q

Flow velocity

FIO

Q- velocity

BFdefects

E

R

U


Landau-de Gennes Nematodynamics

( )

( ) ( ){ }( )( )

( ) ( ){ }

( ) ( )

t * * * * * *1 2

* * * * * *4

* * T *2

1

Er 2 : 3

Er : :

3 2 2 : U 3 33 : :

2UL3 3. . :

U R L

⎛ ⎞⎧ ⎫= ν + ν ⋅ + ⋅ − +⎨ ⎬⎜ ⎟⎩ ⎭⎝ ⎠

⎡ ⎤ν + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ − ⋅ +⎣ ⎦

⎡ ⎤⎧ ⎫− β −β ⋅ + ⋅ − +⎨ ⎬⎢ ⎥⎩ ⎭⎣ ⎦

⎡ ⎤β + ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ − ⋅⎣ ⎦

+ − + −∇ ∇ − ∇

t A Q A A Q Q A I

A Q Q A Q Q Q A Q Q Q A Q Q A I

H H Q Q H H Q I

H Q Q H Q Q Q H Q Q Q H Q Q H I

H Q Q H Q Q ( ) ( )* T ⎡ ⎤

⋅ ⋅ ∇⎢ ⎥⎣ ⎦

Q Q

Erast tttt ++=

Flow velocity

FIO

Q- velocity

BFdefects

s

ˆ c−β⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= ⊗⎜ ⎟ ⎜ ⎟ ⎜ ⎟β⎝ ⎠ ⎝ ⎠⎝ ⎠

a ATHQ


LdG Nematorheology

2* 2 2 2

1 4

22

2

22

3

8 82 ( )9 9 12

1 (2 )3 2

1 (2 )3 2

SS S S

SS S S

SS S S

α η ν β

α η β

α η β

⎛ ⎞= − − +⎜ ⎟

⎝ ⎠⎛ ⎞

= − − + −⎜ ⎟⎝ ⎠⎛ ⎞

= − + −⎜ ⎟⎝ ⎠

2* * * 2 2

4 1 2 42 1 4 (1 )3 3 9 4

SS S Sα η ν ν ν β⎛ ⎞

= − + + − −⎜ ⎟⎝ ⎠

* 2 2 25 2

* 2 2 26 2

1 1 1(2 ) (4 )3 2 31 1 1(2 ) (4 )3 2 3

S S S S S S S

S S S S S S S

α η ν β β

α η ν β β

⎛ ⎞= + + − + − −⎜ ⎟⎝ ⎠⎛ ⎞= − + − + − −⎜ ⎟⎝ ⎠

Trial by Fire

( )S6

SS24 2−+β=λ

( )1 2 3 1 2 1 28 C Cη + η + η = + η − η 2.77<C2<3.84

Hess-Doi models respect reality only if λ<0 for rods!2 *

22 *

2

8 16C4

β + ν=

ν + β( )24 2

06

S SS

βλ

+ −= >

LdG LE


Flow-Modeling


Topological Defects


Point Defects

Brooks-Taylor Spherulites Micron-range Gas Bubbles

bubble-defect dipole:

d=1.2-1.5 R

radial hyperbolic


Disclination Lines in c/c Composites

( 2) / 2S N= − −Zimmer’s Rule:

singular core(3,5,7..)

escape core(4,6,..)

Poincare theorem


Defect-Defect Reactionssplitting: M=+1 2 M=+1/2

annihilation:1/2+(-1/2)=0

asymmetric annihilation with HI

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 00 .1 5

0 .2 0

0 .2 5

0 .3 0

0 .3 5

0 .4 0

0 .4 5

is o tro p ic

( )0.65T = 3 / 8 - H / - 37

1 2M MFd

∝


Defect Rheo-physics:Nucleation1. Frank-Reid loop emission

1λ >

2. Loop splitting

3. Heterogeneous re-orientation

1c

c

N E EE E

∝ − ⇒ ∝−


Defect Rheo-physics: Coarsening Rate f(De)

1. defect-interface annihilation

2. defect-defect annihilation

⋅ 0.2T = a - b Dem

onod

omai

npolydomain






5. Applications


Multiscale Flow-Modeling

Materials Modeling Research Group( )oωτ 1≈o

Lima and Rey , Rheologica Acta (2005), W.R. Burghardyt , JOR (1991)

MONOCRYSTAL LINEAR VISCOELASTICITY( ) 2 3: α α= + +

visco-elasticviscous

τ η n A nN Nn

( ) 12 1 λ

2= − +

γα

( ) ( ) δF-δ

⎧ ⎫= = ⋅ −⎨ ⎬⎩ ⎭

N ω w I - nn A.nn

λ

( )s= ∇A v

2 22

1 33 22

θ,

θ

⎧ = +⎪⎨ ∂

= −⎪ ∂⎩K

y

τ η γ α

θγ α γ

1 32

1 11 32

θ,

θ

⎧ = +⎪⎨ ∂

= −⎪ ∂⎩K

y

τ η γ α

θγ α γn

1ηSPLAY

BEND

n

2η

( ) 13 1 λ

2= −

γα

0.001 0.01 0.1 1 10 100 1000 10000 100000 1000000DIMENSIONLESS FREQUENCY, ( )

1.3

1.4

1.5

1.6

LOSS

TAN

GEN

T, (

)

ω

δ

~

π/2

Loss

Ang

l e

Viscous

Viscoelastic

θ=0 2

2 =0∂∂y

θ

?


Polycrystal Linear Viscoleasticity: G’ and G”

s F d: :dt

δ ⎛ ⎞Δ = − ⎜ ⎟δ ⎝ ⎠nT A

n


Multiscale Flow-Modeling


Monodomain Shear Rheology of CMss

ˆ c−β⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= ⊗⎜ ⎟ ⎜ ⎟ ⎜ ⎟β⎝ ⎠ ⎝ ⎠⎝ ⎠

a ATHQ

theoryexperiments

( )24 21

6 S SS

βλ

+ −= < −


Shear Flow-Alignment and Texturing

Experiments Simulations

transient

steady


Structuring by Flow Through Screens

Experiments

Nucleation of Disclination Latticesn-field RPOM

( ) ( )arg 1/ 4 4 1/ 2 2Ch e = + × + − ×


Fiber Texture EngineeringHigh T Low T

<

Hea

t tra

nsfe

r app

l.

elastic instability

Zig-Zag

ribbon

ribbon

circular

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 00 .1 5

0 .2 0

0 .2 5

0 .3 0

0 .3 5

0 .4 0

0 .4 5

is o tro p ic

R/ξ

( )0.65T = 3 / 8 - H / - 37


Computational Textured NematodynamicsFlow velocity

FIO

Q- velocity

BFdefects

Numerical Scheme

−∇ ⋅ (∇v + ∇vT ) + ∇p + (1− α )∇ ⋅ (G + GT ) = ∇ ⋅τ LCP

where τ LCP = τ total − α(G + GT ) and 0 ≤ α ≤ 1

∇ ⋅ v = 0 G − ∇v = 0


OutlookModels- Need further development in molecular models to remove current

inconsistencies in fitting shear viscosities- Further develop interfacial and contact line nematodynamics- Stress boundary conditions for outflows- Incorporate couple stresses

Computation

- Multiscale multidimensional DNS- Resolve banded texture enigma

Alejandro D. Rey Materials Modeling Research Group ...

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