1. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
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:
TU Berlin Hess et. al.

2. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
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
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

3. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
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
nematic phase
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

4. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Model variables
Orientational
degree of
freedom:
rod-like
disk-like
alignment tensor
symmetric traceless
Flow:
Flow velocity:
Vorticity:
Strain rate:
Pressure tensor:
(stress tensor)
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

5. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
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)
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

6. Landau-de Gennes potential
Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Landau-de Gennes potential
(uniaxial)
nematic phase
isotropic phase
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

7. Constitutive relation for the pressure tensor
Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Constitutive relation for the pressure tensor
momentum balance:
Costitutive
equations:
C. Pereira Borgmeyer and S. Hess J. Non-Equilib. Thermodyn. 20, p. 359 (1995)
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

8. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Complete set of equations
and boundary conditions:
dynamical equation for
the alignment tensor
constitutive equation
momentum balance
equation
Boundary conditions:
strong anchoring
alingment flux
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

9. Plane Couette flow geoemtry
Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Plane Couette flow geoemtry
Effective one dimensional problem:
Scaling:
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

10. Scaled model equations
Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Scaled model equations
Relaxation equation:
Momentum balance equation:
Pressure tensor:
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

11. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Parameters
Scaled second
newtonian viscosity:
Reynold number:
Eriksen number :
Deborah number :
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

12. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
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:
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

13. Apparent slip flow and hydrodyn. jets
Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Apparent slip flow and hydrodyn. jets
Parameters:
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

14. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Apparent slip:
Hydrodynamical jet:
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

15. Orientational dynamics
Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Orientational dynamics
Very low values
of the
order parameter.
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

16. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
The generation of jets
inflection point
The first jet starts at time
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

17. Order parameter vs. hydrodynamical jet
Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Order parameter vs. hydrodynamical jet
Parameters:
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

18. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Definition of different layers I
III II
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Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

19. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Dynamics of the order parameter I II
III I
III II
Orbits of the alignment
tensor components.
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

20. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Angle between the main director and the x-Axes
in the shear plane
wagging
tumbling I
tumbling II
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

21. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Angle between the main director and the x-Axes
in the shear plane
wagging
tumbling I
tumbling II
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

22. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Angle between the main director and the x-Axes
in the shear plane
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

23. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Time where the jet generates and decays.
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

24. Jet generation Frank elasticity 08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers

25. Visualization of the alignment tensor
Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Visualization of the alignment tensor
The length of the edge are given by
the eigenvalues.
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

26. Shear band fluctuations
Parameters:
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers

27. Shear band fluctuations
Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Shear band fluctuations
two
four
The transition between
the bands is driven by
the jets.
three bands
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

28. Shear band fluctuations
Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Shear band fluctuations
odinary
Couette
flow profile
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

29. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

30. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Stable shear bands
Parameters:
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

31. Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Stable shear band flow
Shear band fluctuations
Stable shear band flow
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

32. Orientational dynamics
Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Orientational dynamics
Shear band fluctuations
Stable shear band flow
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.

33. 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.
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers

34. Thank you for your attention!
Microrheology and Rheological Phenomena in Microfluidics
08/12/2018
Slip Stream
Thanks to: Siegfried Hess, Patrick Ilg, Stefan Grandner
Russ Albright
Thank you for your attention!
08/12/2018
Wall-effects on the flow of nematic liquid crystals and liquid crystal polymers
TU Berlin Hess et. al.