Rotational Diffusion and Viscosity of Liquid Crystals E.M. Terentjev M.A. Osipov (~1988) C.J. Chan (~2002) Department of Physics.

Rotational Diffusion and Viscosity

of Liquid Crystals

E.M. Terentjev

M.A. Osipov (~1988) C.J. Chan (~2002)

Department of Physics

Today:

• Summary of the continuum Leslie-Ericksen approach to nematic viscosity: symmetry and basic physical features• 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


• 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

formulationb) eliminating fast variables (velocities) – Smoluchovski

equationc) equilibrium case – spectrum of rotational relaxation modes




• Part 2 –Kinetic theory of rotational diffusion: a) from stochastic Langevin to kinetic Fokker-Planck formulationb) eliminating fast variables (velocities) – Smoluchovski equationc) 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






• Final steps: Leslie coefficients – limits of rod- & disk-like nematic; smectic; experiment

• The missing link: translational part of viscosity…

Leslie-Ericksen continuum

Balance of local forces (stress) and local torques (nematic):

0)(

)(

)(

)(

n

ST

n

L

0)(

)(

)(

)(

u

ST

u

L

],,,[221 EnnFdL uux

],[ ux nRdST Lagrangian

Entropy production

Potential energy: Frank elasticity Order-parameter expansionSmectic (n-layer coupling)Gels (n-elastic matrix coupling)

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)

)])([()])([(

)]([4][2)(

22

121

25

24

21

nnn

nnAnnAnnAR

dtd

dtd

Linear nematic viscosity (Leslie-Ericksen): strain rates + n-rotation rates

Here: DT )(21 uu

W ucurl21 ][ nn Relative to

)])([()])([(

)]([4][2)(

22

121

25

24

21

nnn

nnAnnAnnAR

dtd

dtd

)])([()]([

)]([4][2)(

22

121

25

24

21

nnDnD

nnCnnCnnCF

Viscous flow:

Elastic deformation:

Symmetry: 3 flow geometries

G. D. V.

n nn

)(

)(

54221

64321

421

c

b

a

)2(

)(

65481

5

441

4

654121

1

A

A

A

221 )( n KFFrank elasticity: Penalises local torques… balanced by boundaries

or by viscous torques (antisymmetric visc.stress)

Balance of forces:

Balance of torques:

elastvisc)( uuu

Assuming , non-dimensional number: K

vL

nK

v ~

)(

)(r

2

E

)])([()])([(

)]([4][2)(

22

121

25

24

21

nDnn

nDnAnDnAnDnAR

dtd

dtd

Viscous flow:

)()]([][ 212 DKI dt

ddtd nωΩnnnn

Anisotropic continuum with linear friction

Improvements• Qij theory (Sonnet & Virga)• Flow with n-gradients

• Compressible medium (acoustics)• non-Newtonian (beyond G*=C+iA)

>>1

Microscopic Theory of Viscosity

General scheme: 1) Determine the Microscopic Stress tensor, M, from local molecular dynamics2) Identify the kinetic (Fokker-Planck) equation for ensemble in flow3) Solve is to find the (non-equilibrium) molecular distribution function4) The answer:

Macroscopic (continuum) stress tensor

vdvP ),()(σM

What about classical isotropic liquids?

ijijjiij Pvv

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; This requires the full pair-correlation

function description of molecular distribution.

Bits of history: 1975-1982 Diogo-Martins 1978 Tsebers 1982 Marucci

1980-1983 Kuzuu-Doi

1989-1991 Osipov-TerentjevChrzhanovska-SokalskiFialkowski Kroger1990-1995 S.T. Wu (expt)

MarucciSemenov

Larson

Microscopic Stress tensor

a

“Number” density

)()(

rxx

r

),()()()( 2 rrxrrxrrx Ox

…formally expanding in powers of r

r

r

N

1

)()(

rrxx

N

m1

)(][)(

rrxrωvxp

Momentum (rigid body rotation)

Mσ)(

dt

d xpDefinition:

)()(][)(][)(1

rrxrrrωvrrxrωrωvxp

mmN

…evaluating the t-derivative

…we obtain the translational and the orientational parts of M:

)()]()()()([

)()][)(][σ

21112

''

'21M

rxIΓIΓIIωIωωIωI

rxFrrxvv

Tr

m

Uniaxial particles; Uniaxial mean-field

kiikik aaIII )( || Ia

a

Ellipsoid: p=a/b2

52

||22

51 );( mbIbamI

),,()( '''

aarrrΓ

Ua

ak

jijkiAlso: the torque

Finally (only orientational part):

)(]1

1

1

1

1

))((1

1[σ

2

2

22

2

22

2M

rxa

aaaaaa

a

aaaωωaaωaω

U

p

pU

p

U

p

p

Ip

p

Averaging over (fast) angular velocity distribution gives the “kinetic” part of M

(which is the momentum flux due to motion of molecules, in contrast to the “potential” part of M representing the flux due to intermolecular forces.

)()(1

13 3

12

2

rxδaa

p

pTkB

In a dilute gas, e.g. a solution of rod-polymers, the kinetic part, M=3kBT(aa1/3), is the main contribution; but in a dense molecular liquid – the potential part is dominant.

Cylinder: p=a/b2

21

||22

121 );3( mbIbamI

Equilibrium is, of course:

Tk

U

BeP)( na

Non-equilibrium distribution functionMean-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 avavaaΩ )curl(curl)( 2

121

1

121

2

2

D

p

p

12

ITkB

a Tk

D BrotSeparation 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.

]))(curl[(])[(

)()(

||22

2

PI

IP

PTk

PPP

aaaa

Baaat

aΩva

ΓΩ

)( na MFaU

From stochastic (Langevin) to kinetic (F-P)

Full angular velocity of uniaxial object: ωaΨ ψ ωaL II IIψ

Equation of (rotational) motion: ΓΩωaωaψaL )ψ(ψ |||| III

)ψ(ψ

)()(),,,ψ( P-P-P-tPt

ωω

aa

aω

PI

PI

aξξaaaξξaaωω 22

2

2||

2 1

ψ)()(

1

aaξξ TkTk BB )(22 ||

)]([)( PI

PP at

Γaa

ωω

PI

TkP

IP

IP

I

TkP

IBB

ψ2

1)curl-ψ(

ψ)(ψ)(

2

1)(

||21

||

||||

vaaωaa

ωΩω

ω

Arrive at full F-P equation for rotational motion:

Eliminating fast variables Integrating out and

ψ ω

1) Ignore the relaxation of (too fast):ψ

),,(~

221 )curl-ψ(

2 tPeP Tk

I

B

II

ωava

Put in and integrate over ψ

),(~

2)-(2 tPeP Tk

I

B aΩω

Naively: and integrate over

2) Ignore the relaxation of (faster than the coordinate a):ω

ω )( PP at Ω (?!)

Substitute to F-P, only retain leading terms in <<1 and expand in powers of small deviation , matching the terms of the same order gives Y(a, ). THEN integrate over …..ω

)]([)( PI

PP at

Γaa

ωω

P(I

PI

TkP

IB )()curl

2)(

2

1)( || aωvaaa

ωΩω

ω

Diffusive corrections to the coordinate-only kinetic equation arise from the last bits of non-relaxed Maxwell distribution:

])-[,()(),(~

2)-(2 Ωωaa

Ωω

YtPePa

Tk

I

B

Solving the F-P equation

)( na MFaU

])[(]))(curl[( 2||2 PPI

Iaaa ΩΩaΩva

Spectrum of rotational relaxation modes: eigenfunction expansion in equilibrium fluid

),()( tPPt aa )()(1

0 aa n

t

nn wecwP n

)][()( 12 PUPPP MFaTkaaat B Ω

n aTkU BMFew /)(

0 )( aLook for solutions in the form… then

)()()( 0 aa wfw nn

nn

nMF

B

n ffU

Tk

f

22

2 11cot

nTk

U

Tk

Uf

ee BB sinsin

1)()(

Gives self-consistent integral equation for wn

0 0

)(

2

)(

sin)(sin

11)(

x

nTk

xU

n

Tk

U

n dzzzwdxex

Cew BB

2sin)( JSMFU na

Equilibrium relaxation time(s)

Could only solve by expansion to leading order in the smallest non-zero eigenvalue [] (i.e. the longest relaxation time ):

0 0

)()(

12

)(

sinsin

111)(

x Tk

zU

Tk

xU

Tk

U

n dzzedxex

ew BBB

...11

)( 2

2

121

120

wwwwn

Condition of periodicity states: w()=w()=const (of order 1 for normalized w) gives:

1sinsin

110 0

)()(

12

x Tk

zU

Tk

xU

dzzedxex

BB

The longest relaxation time [of diffusion over the orientational barrier U() ]:

2sinJSUMF

0 0

)()(

1 sinsin

1 x Tk

zU

Tk

xU

B

dzzedxexTk

BB Take limits: S0 and S1

2sinJSUMF

Felderhof, and many others… (1989-now)

...)1(1 Tk

JS

Tk BB

S0 Tk

JS

B

B

BeJS

Tk

Tk

2/3

1

S1

Solving the non-equilibrium F-P eq.

In order to calculate the ensemble average of microscopic stress tensor M in a system with flow gradients, we need to find the correction to equilibrium pdf

P(a,,t) }{)()(3eq.3

1 pressureconstTkB

rxδaa

}{0)()(eq.

symmetryU

rxnaa a

])[(]))(curl[( 2||2 PPI

Iaaa ΩΩaΩva


It turns out that there is no need to average the symmetric part of M (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 ~ v).

Symmetric part of <M>: ),(

~21

31

2

2

sym

M 2)(31

1σ

tP

B

UUUTk

p

p

aa

aaaaaa

aδaa


)( 31 δaa Multiply by …and integrate da

WWDDDp

p

tSHL

aaaaaaaaaaaaδaa 21

2

2

21

31 :2

1

1...

WWDDDp

p

tSHL

aaaaaaaaaaaaδaa 21

2

2

21

31 :2

1

1...

UUUTk

SHRB

aaa aaaaaδaa 21

6... 31

aaaaaav ;;(1

12

2

2symGQ

tp

pTkBM

WWDDDp

p

tSHL

aaaaaaaaaaaaδaa 21

2

2

21

31 :2

1

1...

Solving the non-equilibrium F-P eq.


To average the antisymmetric part of M , describing the torques, we may only consider the t-dependent rotation of n (without any ~ v).

)][(0 eq1

eq2 PUP MFaTkaa B

);(1)(eq tYPP na,na,

212

23 )( naJSUMF

]1)[)((3

][2

12 YTk

JSYUY

BaaTka B

nana

Using the fact that (na) is negligibly small.

Thus find ))(()/(2

)/(nana

TkJS

TkJS

TkY

B

B

B

And the average aaa dYeUU Tk

U

aaM B

21

antisym

][)1257()/(2

)/(4

2

701 nnnn

SS

TkJS

TkJS

B

B

Leslie coefficients, and consequences

)43(

)43(

257

][

1257][

41

171

1

121

6

41

171

1

121

5

4

2

1

1351

4

1

1211

121

3

4/2/

351

111

121

2

4

2

1

11

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

SSS

SSS

SS

SggS

SSggS

S

p

p

p

p

p

p

p

p

p

p

p

p

p

p

kTJSkTJS

p

p

p

p

An issue with isotropic Leslie coefficient 4An issue with the “Arrhenius” activation energy and dependenceRod-like (p>>1) or disk-like (p<<1) nematic liquidsTorque balance not always possible: director rotation vs. tumbling

1

1

2

3

...)(...

...)(...tan

gp

gp

Rods (p>>1) give tan ~ 1;

~45o

Disks (p<<1) suggest tan>>1; ~90o

Conclusions

• 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)<<1

• Leslie coefficients expressed either as an expansion in powers of {S}, or as

Arrhenius-style activation exponentials {exp –D/kT}

• What about inhomogeneous particle density xrx (i.e. in mixtures)?

• How to solve for the full spectrum of relaxation modes (including coupling)?

• Non-axisymmetric particles (the derivation depends on general ik)?

• 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)<<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 4 (and all of the

residual isotropic viscosity!..) is a totally different story…

•What about inhomogeneous particle density xrx (i.e. in mixtures)?

• How to solve for the full spectrum of relaxation modes (including coupling)?• Non-axisymmetric particles (the derivation depends on general ik)?• 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)?

Thank you

Rotational Diffusion and Viscosity of Liquid Crystals E.M. Terentjev M.A. Osipov (~1988) C.J. Chan (~2002) Department of Physics.

Documents

general principles

nematic viscosity

available options summary

basic physical features

cases summary

stochastic langevin

viscosity of liquid

department of physics