alejandro d. rey materials modeling research group
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Materials Modeling Research Group
Flow and Texture Modeling of Liquid Crystalline Materials
Alejandro D. Rey
Materials Modeling Research Group
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
Materials Modeling Research Group
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
Materials Modeling Research Group
Liquid Crystalline Precursors for Structural Materials
Materials Modeling Research Group
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
Materials Modeling Research Group
Outline1. Introduction: Liquid crystals
2. Modeling Overview
3. Leslie-Ericksen Nematodynamics
4. Landau de Gennes Nematodynamics
5. Applications
Materials Modeling Research Group
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
Materials Modeling Research Group
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 ⎝ ⎠
Materials Modeling Research Group
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 >
Materials Modeling Research Group
Outline1. Introduction: Liquid crystals
2. Modeling Overview
3. Leslie-Ericksen Nematodynamics
4. Landau de Gennes Nematodynamics
5. Applications
Materials Modeling Research Group
Flow-Modeling
nQ
Materials Modeling Research Group
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
Materials Modeling Research Group
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
Materials Modeling Research Group
• Flow alignment
• Shear viscosities
• Back-flow
• Secondary flow
• First normal stress difference
• Shear thinning
Leslie-Ericksen Nematorheology
Materials Modeling Research Group
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
Materials Modeling Research Group
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
Materials Modeling Research Group
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
Materials Modeling Research Group
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
Materials Modeling Research Group
Flow-Modeling
Materials Modeling Research Group
Topological Defects
Materials Modeling Research Group
Point Defects
Brooks-Taylor Spherulites Micron-range Gas Bubbles
bubble-defect dipole:
d=1.2-1.5 R
radial hyperbolic
Materials Modeling Research Group
Disclination Lines in c/c Composites
( 2) / 2S N= − −Zimmer’s Rule:
singular core(3,5,7..)
escape core(4,6,..)
Poincare theorem
Materials Modeling Research Group
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
∝
Materials Modeling Research Group
Defect Rheo-physics:Nucleation1. Frank-Reid loop emission
1λ >
2. Loop splitting
3. Heterogeneous re-orientation
1c
c
N E EE E
∝ − ⇒ ∝−
Materials Modeling Research Group
Defect Rheo-physics: Coarsening Rate f(De)
1. defect-interface annihilation
2. defect-defect annihilation
⋅ 0.2T = a - b Dem
onod
omai
npolydomain
Materials Modeling Research Group
Outline1. Introduction: Liquid crystals
2. Modeling Overview
3. Leslie-Ericksen Nematodynamics
4. Landau de Gennes Nematodynamics
5. Applications
Materials Modeling Research Group
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
?
Materials Modeling Research Group
Polycrystal Linear Viscoleasticity: G’ and G”
s F d: :dt
δ ⎛ ⎞Δ = − ⎜ ⎟δ ⎝ ⎠nT A
n
Materials Modeling Research Group
Multiscale Flow-Modeling
Materials Modeling Research Group
Monodomain Shear Rheology of CMss
ˆ c−β⎛ ⎞ ⎛ ⎞ ⎛ ⎞
= ⊗⎜ ⎟ ⎜ ⎟ ⎜ ⎟β⎝ ⎠ ⎝ ⎠⎝ ⎠
a ATHQ
theoryexperiments
( )24 21
6 S SS
βλ
+ −= < −
Materials Modeling Research Group
Shear Flow-Alignment and Texturing
Experiments Simulations
transient
steady
Materials Modeling Research Group
Structuring by Flow Through Screens
Experiments
Nucleation of Disclination Latticesn-field RPOM
( ) ( )arg 1/ 4 4 1/ 2 2Ch e = + × + − ×
Materials Modeling Research Group
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
Materials Modeling Research Group
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
Materials Modeling Research Group
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
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