1. Rotational Diffusion and Viscosity of Liquid Crystals
Department of Physics
Rotational Diffusion and Viscosity of Liquid Crystals
E.M. Terentjev
M.A. Osipov (~1988) C.J. Chan (~2002)
I would like to talk today about some unusual morphologies in a liquid crystal colloid. The main difference between the regular colloids and liquid crystal colloids is the fact that the suspending liquid has a broken symmetry and associated elastic energy of deformations. This means that whenever we put particles in such a liquid, they will do two things:
1. They will tend to distort the LC order and may produce topological defects in the matrix.
2. There will be new type of forces between them which are elastic in character and are mediated by the anisotropic liquid matrix. No such forces exist in the ordinary colloidal systems.
These two crucial points lead to number of novel phenomena concerning the phase behaviour and morphology in liquid crystalline colloids. I would like today to talk about one such system.

2. Today: Building the Microscopic theory of continuum linear response:
Summary of the continuum Leslie-Ericksen approach to nematic viscosity: symmetry and basic physical features
Building the Microscopic theory of continuum linear response:
general principles and available options
Part 1 – Mean-field Microscopic Stress Tensor, from first principles,
examine rod-like and disk-like limiting cases
Part 2 –Kinetic theory of rotational diffusion:
a) from stochastic Langevin to kinetic Fokker-Planck formulation
b) eliminating fast variables (velocities) – Smoluchovski equation
c) equilibrium case – spectrum of rotational relaxation modes
Part 3 – Solving non-equilibrium kinetic equation:
a) the “Doi trick”
b) antisymmetric stress tensor
Final steps: Leslie coefficients – limits of rod- & disk-like nematic;
smectics; experiment
Summary of the continuum Leslie-Ericksen approach to nematic viscosity: symmetry and basic physical features
Building the Microscopic theory of continuum linear response:
general principles and available options
Part 1 – Mean-field Microscopic Stress Tensor, from first principles,
examine rod-like and disk-like limiting cases
Part 2 –Kinetic theory of rotational diffusion:
a) from stochastic Langevin to kinetic Fokker-Planck formulation
b) eliminating fast variables (velocities) – Smoluchovski equation
c) equilibrium case – spectrum of rotational relaxation modes
Part 3 – Solving non-equilibrium kinetic equation:
a) the “Doi trick”
b) antisymmetric stress tensor
Final steps: Leslie coefficients – limits of rod- & disk-like nematic;
smectic; experiment
The missing link: translational part of viscosity…
Summary of the continuum Leslie-Ericksen approach to nematic viscosity: symmetry and basic physical features
Building the Microscopic theory of continuum linear response:
general principles and available options
Part 1 – Mean-field Microscopic Stress Tensor, from first principles,
examine rod-like and disk-like limiting cases
Part 2 –Kinetic theory of rotational diffusion:
a) from stochastic Langevin to kinetic Fokker-Planck formulation
b) eliminating fast variables (velocities) – Smoluchovski equation
c) equilibrium case – spectrum of rotational relaxation modes
Part 3 – Solving non-equilibrium kinetic equation:
a) the “Doi trick”
b) antisymmetric stress tensor
Summary of the continuum Leslie-Ericksen approach to nematic viscosity: symmetry and basic physical features
Building the Microscopic theory of continuum linear response:
general principles and available options
Part 1 – Mean-field Microscopic Stress Tensor, from first principles,
examine rod-like and disk-like limiting cases
Summary of the continuum Leslie-Ericksen approach to nematic viscosity: symmetry and basic physical features
Building the Microscopic theory of continuum linear response:
general principles and available options
Part 1 – Mean-field Microscopic Stress Tensor, from first principles,
examine rod-like and disk-like limiting cases
Part 2 –Kinetic theory of rotational diffusion:
a) from stochastic Langevin to kinetic Fokker-Planck formulation
b) eliminating fast variables (velocities) – Smoluchovski equation
c) equilibrium case – spectrum of rotational relaxation modes
Summary of the continuum Leslie-Ericksen approach to nematic viscosity: symmetry and basic physical features
Building the Microscopic theory of continuum linear response:
general principles and available options
Summary of the continuum Leslie-Ericksen approach to nematic viscosity: symmetry and basic physical features

3. Leslie-Ericksen continuum
Balance of local forces (stress) and local torques (nematic):
Lagrangian
Potential energy: Frank elasticity
Order-parameter expansion
Smectic (n-layer coupling)
Gels (n-elastic matrix coupling)
Entropy production
Dissipation function: friction arising from relative motion of fluid and internal variable n
(Q: symmetry criteria?.. A: assign a dissipation to each principal deformation mode)
Linear nematic viscosity (Leslie-Ericksen): strain rates + n-rotation rates
Here:
In a liquid (unless we talk about gels and smectics) the potential energy does not depend on deformation u
But the dissipation function does!
Relative to

4. Symmetry: 3 flow geometries
Viscous flow:
Elastic deformation:

5. Anisotropic continuum with linear friction
Viscous flow:
Frank elasticity:
Penalises local torques… balanced by boundaries
or by viscous torques (antisymmetric visc.stress)
Improvements
Qij theory (Sonnet & Virga)
Flow with n-gradients
Compressible medium (acoustics)
non-Newtonian (beyond G*=C+iwA)
>>1
Balance of forces:
Balance of torques:
Assuming g1~g2, non-dimensional number:

6. Microscopic Theory of Viscosity
General scheme:
Determine the Microscopic Stress tensor, sM, from local molecular dynamics
Identify the kinetic (Fokker-Planck) equation for ensemble in flow
Solve is to find the (non-equilibrium) molecular distribution function
The answer:
Macroscopic (continuum) stress tensor s =
Bits of history: Diogo-Martins
1978 Tsebers
1982 Marucci
Kuzuu-Doi
Osipov-Terentjev
Chrzhanovska-Sokalski
Fialkowski
Kroger
S.T. Wu (expt)
Marucci
Semenov
Larson
What about classical isotropic liquids?
This requires a lot more complicated theory than is possible in liquid crystals, where the orientational mean field allows to capture all anisotropic effects – and relative values of all Leslie coefficients (if nematic).
However, no mean-field theory will ever determine the isotropic viscosity: the only constant that would survive if Q=0; a4h. This requires the full pair-correlation function description of molecular distribution.

8. Microscopic Stress tensor
“Number” density ra
rab
Momentum (rigid body rotation)
Definition: aa ra
…evaluating the t-derivative
…formally expanding in powers of rab
…we obtain the translational and the orientational parts of sM:

9. Uniaxial particles; Uniaxial mean-field
Ellipsoid: p=a/b
Cylinder: p=a/b
Also: the torque
Finally (only orientational part):
Averaging over (fast) angular velocity distribution gives the “kinetic” part of sM
(which is the momentum flux due to motion of molecules, in contrast to the
“potential” part of sM representing the flux due to intermolecular forces.
In a dilute gas, e.g. a solution of rod-polymers, the kinetic part, sM=S3kBT(aa-1/3), is the main contribution; but in a dense molecular liquid – the potential part is dominant.

10. Non-equilibrium distribution function
Mean-field kinetic theory
Full Fokker-Planck description with sources (velocity gradients) is reduced due
to two-step relaxation feature: the distribution of velocities relaxes much faster
to the Maxwell-like form with the mean
Separation of time scales is assured:
The reduced Smoluchowski equation (for pdf only dependent on coordinates)
relaxes much slower to the steady state; dissipation and effective friction forces
during this relaxation is the main source of viscous response in dense liquids.
Equilibrium is, of course:

11. From stochastic (Langevin) to kinetic (F-P)
Full angular velocity of uniaxial object:
Equation of (rotational) motion:
Arrive at full F-P equation for rotational motion:

12. Eliminating fast variables
Integrating out and
1) Ignore the relaxation of (too fast):
Put in
and integrate over
Naively:
and integrate over
2) Ignore the relaxation of (faster than the coordinate a):
(?!)
Diffusive corrections to the coordinate-only kinetic equation arise from the last bits
of non-relaxed Maxwell distribution:
Substitute to F-P, only retain leading terms in a<<1 and expand in powers of small deviation (w-W), matching the terms of the same order gives Y(a, [w-W]).
THEN integrate over …..

13. Solving the F-P equation
Spectrum of rotational relaxation modes: eigenfunction expansion in equilibrium fluid q n a
Look for solutions in the form
… then
Gives self-consistent integral equation for wn

14. Equilibrium relaxation time(s)
Could only solve by expansion to leading order in the smallest non-zero eigenvalue [1/t1]
(i.e. the longest relaxation time t1):
Condition of periodicity states: w(q=p)=w(0)=const (of order 1 for normalized w) gives:
The longest relaxation time [of diffusion over the orientational barrier U(q) ]:
Take limits: S0 and S1
Felderhof, and many others… (1989-now)
S0
S1

15. Solving the non-equilibrium F-P eq.
In order to calculate the ensemble average of microscopic stress tensor sM in a system with flow gradients, we need to find the correction to equilibrium pdf P(a,w,t)
It turns out that there is no need to average the symmetric part of sM (there is
a “trick” allowing its direct evaluation). To average the antisymmetric part, describing
the torques, we may only consider the t-dependent rotation of n (without any W ~ v).
Symmetric part of <sM>:
Multiply by
…and integrate da

16. Solving the non-equilibrium F-P eq.
To average the antisymmetric part of sM , describing the torques, we may only
consider the t-dependent rotation of n (without any W ~ v).
Using the fact that (na) is negligibly small .
Thus find
And the average

17. Leslie coefficients, and consequences
An issue with the “Arrhenius” activation energy and dependence
Rod-like (p>>1) or disk-like (p<<1) nematic liquids
Torque balance not always possible: director rotation vs. tumbling
Rods (p>>1) give tanq ~ 1; q~45o
Disks (p<<1) suggest tanq>>1; q~90o
An issue with isotropic Leslie coefficient a4

18. Conclusions
What about inhomogeneous particle density r(x)=S d(ra-x) (i.e. in mixtures)?
How to solve for the full spectrum of relaxation modes (including q-f coupling)?
Non-axisymmetric particles (the derivation depends on general Iik)?
Non-axisymmetric mean-field U(a•n) (e.g. biaxial phases, or smectics)?
Could one re-formulate the whole theory in terms of Q-tensor?
Moving away from the mean-field approximation (pair correlation functuions)?
First-principle derivation of the microscopic stress tensor
Derivation of full F-P equation allows any level of accuracy, as long as the
corresponding HD problem is resolved
Separation of time scales gives the relevant (Smoluchowski) equation, with
the hierarchy of sources in powers of a=(tw /ta)<<1
Leslie coefficients expressed either as an expansion in powers of {S}, or as
Arrhenius-style activation exponentials {exp –D/kT}
First-principle derivation of the microscopic stress tensor
Derivation of full F-P equation allows any level of accuracy, as long as the
corresponding HD problem is resolved
Separation of time scales gives the relevant (Smoluchowski) equation, with
the hierarchy of sources in powers of a=(tw /ta)<<1
Leslie coefficients expressed either as an expansion in powers of {S}, or as
Arrhenius-style activation exponentials {exp –D/kT}
The translational part of viscous coefficients, giving a part of a4 (and all of the
residual isotropic viscosity!..) is a totally different story…
What about inhomogeneous particle density r(x)=S d(ra-x) (i.e. in mixtures)?
How to solve for the full spectrum of relaxation modes (including q-f coupling)?
Non-axisymmetric particles (the derivation depends on general Iik)?
Non-axisymmetric mean-field U(a•n) (e.g. biaxial phases, or smectics)?
Could one re-formulate the whole theory in terms of Q-tensor?
Moving away from the mean-field approximation (pair correlation functions)?

19. Thank you