Wall-effects on the flow of
nematic liquid crystals and
liquid crystal polymers
Sebastian Heidenreich,
Patrick Ilg,
and Siegfried Hess;
Institute for Theoretical Physics, TU Berlin;
Institute for Polymer Physics, ETH Zuerich
contact: sebastian@itp.physik.tu-berlin.de;
s.hess@physik.tu-berlin.de
S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Overview
Model equations:
Relaxation equations for the alignment
tensor
Hydrodynamic equations and constitutive
equations for the pressure tensor
Scaled model equations
Flow effects of tumbling nematics:
Apparent slip flow; hydrodynamical jets
Shear band flow and fluctuations
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Wall-effects on the flow of nematic liquid crystals
and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
For fluids consisting of
nano-rods the orientation
couples with the flow.
Phases:
The shape of molecules
is modeled by
hard rods
orientation
isotropic phase
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nematic phase
Wall-effects on the flow of nematic liquid crystals
and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Model variables
Orientational
degree of
freedom:
disk-like
rod-like
alignment tensor
symmetric traceless
Flow:
Flow velocity:
Strain rate:
Vorticity:
Pressure tensor:
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(stress tensor)
Wall-effects on the flow of nematic liquid crystals
and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Model equations
Dynamical equations for the alignment tensor:
Alignment flux tensor:
... diffusion
coefficient
Derivative of the
model potential:
S. Hess, Z. Naturforsch. 30a, p. 728 (1975);
C. Pereira Borgmeyer and S. Hess J. Non-Equilib. Thermodyn. 20, p. 359 (1995)
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Landau-de Gennes potential
(uniaxial)
nematic phase
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isotropic phase
Wall-effects on the flow of nematic liquid crystals
and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Constitutive relation for the
pressure tensor
momentum balance:
Costitutive
equations:
C. Pereira Borgmeyer and S. Hess J. Non-Equilib. Thermodyn. 20, p. 359 (1995)
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Complete set of equations
and boundary conditions:
constitutive equation
dynamical equation for
the alignment tensor
momentum balance
equation
Boundary conditions:
strong anchoring
alingment flux
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Plane Couette flow geoemtry
Effective one dimensional problem:
Scaling:
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Scaled model equations
Relaxation equation:
Momentum balance equation:
Pressure tensor:
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Parameters
Scaled second
newtonian viscosity:
Reynold number:
Eriksen number :
Deborah number :
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Numerical results
• Explicit finite difference scheme of 4th/3th order
(400 grid points)
• Time integration with a 4th order adaptive Runge
Kutta algorithm
• Nematic phase
• No alignment flux
• Low Reynold number
•
• Boundary conditions:
Strong anchoring:
No slip condition:
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Apparent slip flow and hydrodyn. jets
Parameters:
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Apparent slip:
Hydrodynamical jet:
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Orientational dynamics
Very low values
of the
order parameter.
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
The generation of jets
a units of a K
y units of L
1.0
1.41
0.75
0.5
0.25
0.58
inflection point
0
40
50
60
L
t units of w
v
The first jet starts at time
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Order parameter vs.
hydrodynamical jet
Parameters:
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Definition of different layers
III
II
I
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and liquid crystal polymers
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Dynamics of the order parameter
I
II
III
III
II
I
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Orbits of the alignment
tensor components.
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Angle between the main director and the x-Axes
in the shear plane
wagging
tumbling I
tumbling II
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Angle between the main director and the x-Axes
in the shear plane
wagging
tumbling I
tumbling II
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Angle between the main director and the x-Axes
in the shear plane
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Time where the jet
generates and decays.
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Jet generation
Frank elasticity
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Visualization of the alignment tensor
The length of the edge are given by
the eigenvalues.
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Shear band fluctuations
Parameters:
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and liquid crystal polymers
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Shear band
fluctuations
two
four
The transition between
the bands is driven by
the jets.
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Wall-effects on the flow of nematic liquid crystals
and liquid crystal polymers
three bands
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Shear band fluctuations
odinary
Couette
flow profile
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Stable shear bands
v units of vw
1.0
y units of L
0.75
1
0.5
0
0.25
0
30
300
600
t units of
900
1200
L
vw
Parameters:
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Stable shear band flow
v units of vw
1.0
y units of L
0.75
1
0.5
0
0.25
0
30
300
600
t units of
Shear band fluctuations
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900
1200
L
vw
Stable shear band flow
Wall-effects on the flow of nematic liquid crystals
and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Orientational dynamics
a units of aK
1.0
y units of L
0.75
1
0.5
0
0.25
0
30
300
600
900
t units of
Shear band fluctuations
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1200
L
vw
Stable shear band flow
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Conclusions
Hydrodynamic jets emerge as a
consequence of the interplay of the Frank
elasticity, Wall-effects and the flow
coupling.
Shear band fluctuations are driven by
hydrodynamic jets
For small Frank elasticity constant and
high shear the shear bands are stable.
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and liquid crystal polymers
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S. Heidenreich, P. Ilg and S. Hess, Sfb 448
Thanks to:
Slip Stream
Siegfried Hess,
Patrick Ilg,
Stefan Grandner
v units of vw
1.0
y units of L
0.75
1
0.5
0.25
0
0
30
130
230
t units of
330
L
vw
430
Russ Albright
Thank you for your attention!
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and liquid crystal polymers
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