No optics, no photonics Peter Palffy-Muhoray Liquid Crystal - - PowerPoint PPT Presentation

No optics, no photonics

Peter Palffy-Muhoray Liquid Crystal Institute Kent State University

2/28/2018 1

Light-matter interactions

• light can produce mechanical stress, and do work
• light driven machines:

2/28/2018 2

Light-matter interactions

• light driven machines:
• light

– changes degree of order – change in order parameter produces stress – system does work

• equation of state relates pressure and order parameter
• illuminates the connection of stress and order parameter

2/28/2018 3

Towards an equation of state for dense nematics with steric interactions

Peter Palffy-Muhoray Liquid Crystal Institute Kent State University

2/28/2018 4

collaborators: Eduardo Nascimento Physics, Univ. Sao Paulo Jamie Taylor Mathematics, Oxford, Kent State Epifanio Virga

• Dept. Mathematics, Univ. Pavia

Xiaoyu Zheng

• Dept. of Math. Sciences, Kent State

Outline

• motivation
• system under consideration
• free energy
• the probability distribution function & order parameters
• summary
• outstanding questions

2/28/2018 5

2/28/2018 6

motivation

Motivation

• interparticle interactions typically consist of

– long-range attraction – short range repulsion

• in the case of liquid crystals,

– much is known about long range attraction  – much less is known about short range repulsion ? ?

• we are interested in the behavior of nematics at high

densities where short range steric effects dominate.

2/28/2018 7

Current landscape

• Onsager theory:

– ‘The effects of shape…’, ANYAS 1949 – seminal work providing model of interacting hard rods – predicts

• nematic order above critical density
• phase separation
• simulations (Frenkel,..) and experiments (Lekkerkerker,.)

show

– nematic order – phase separation

2/28/2018 8

MC study of hard ellipsoids

9

Gerardo Odriozola, J. Chem. Phys. 136, 134505, (2012)

2/28/2018

Onsager Theory

• free energy
• where

is the pair excluded volume. for hard spheres: Question: what is the region of validity?

2/28/2018 10

2

1 [ ln( ) ..] 2

exc

V N F kT N V N V    

12

( ) 3 12

( 1)

w kT exc

V e d



  



r

r

3

4 8 8 3

exc

V r v   

Onsager Theory: region of validity

• key step in derivation:

– (ANYAS Eq. 19 – 21)

• but

!?!

• Onsager makes no comment on this

– refers to work of Mayer – suggests region of validity

• .

2/28/2018 11

2 2

ln(1 ..) 2 2

exc exc

N N V V V V   

2 18

10 2

exc

N V V  0.2 Nv V 

Onsager Theory: region of validity

• expansion of the free energy in powers of the density

converges*

• the region of convergence is still not firmly established
• it seems likely that Onsager’s original estimate is

reasonable**

• Onsager theory is valid for dilute systems. We look

elsewhere.

2/28/2018 12

* Lebowitz and Penrose, J. Math. Phys. 7, 841 (1964) ** P. P-M., E.G. Virga, X. Zheng, “Onsager’s missing steps retraced”, J. Phys. Condens. Matter 29 475102 (2017)

Motivation

• current descriptions are low density approximations
• want to describe, even if very approximately, what

happens to orientationally ordered hard particle systems at high densities

• can we approximate what happens when we run out of

available space/available states?

2/28/2018 13

Guidance: Equations of state

• describe bulk behavior

– ideal gas: – Problem: for a given , can be arbitrarily large! – neo-Hookean elasticity: – Problem: stretch can be arbitrarily large!

2/28/2018 14

PV NkT 

• pressure
• volume
• no. of particles
• Boltzmann's constant
• temperature

P V N k T

N V

2 11

1 ( ) G     

11 - stress

• shear modulus
• stretch

G  

Guidance: Equations of state

• describe bulk behavior

– ideal gas: – Problem: for a given , can be arbitrarily large! – Non-ideal: Van der Waals – neo-hookean elasticity: – Problem: stretch can be arbitrarily large! – Non-ideal: Gent

2/28/2018 15

PV NkT 

• pressure
• volume
• no. of particles
• Boltzmann's constant
• temperature

P V N k T

N V

2 11

1 ( ) G     

11 2 2 2 1 1 2 3

• stress
• shear modulus
• stretch

I G        

2

(P a) (V )

• NkT

Nv    

2 11 1

3 1 ( )( )

m m

I G I I       

Origins of the ‘hard’ response

• VdW free energy
• Gent energy density
• Helmholtz free energy density

– free energy as available phase space .

2/28/2018 16

2

1 1 ln( ) 2

• v

a kT v         F

1

1 3 (I 3)ln(1 ) 2 3

m m

I W G I       ln kT Z V   F

  F

Z 

• must keep logarithm for ‘hard’ response!

2/28/2018 17

system under consideration

System under consideration

• assembly of identical hard ellipsoids
• length width .
• cannot interpenetrate
• pair excluded volume
• volume one particle makes unavailable to the center of the
• ther
• and are known functions of the aspect ratio

2/28/2018 18

L W

12 2 12

( ) (cos )

exc

V C DP   

12

 2a 2b

L W

4 ( 4 ) 3

• L

W C v W L    4 ( 2 ) 3

• L

W D v W L   

C D / L W  

Excluded volume

• ellipsoids
• isotropic average:
• parallel configuration:

2/28/2018 19

12

 2a 2b

L W

12 2 12 2 12 12

( ) (cos ) 1 ( ) (3cos 1) 2

exc exc

V C DP V C D         

12

( )

exc

V C    (0)

exc

V C D  

2/28/2018 20

the free energy

Single component system

• Helmholtz free energy
• partition function:
• configurational partition function:

(integrate out momenta)

• pairwise interactions:

2/28/2018 21

ln F kT   Z

( , )

...

kT

d d e



 

q p

q p Z

H ( )

....

U kT

Z e d



 

q

q

( )

1 ln ... ... !

Uij i j kT

N m

F kT e dq dq N



 



q q

Interaction potential

• ignore attractive interactions, keep only hard core

2/28/2018 22 ( ) , ,

1

1 ln ... ... !

R U i j i j i j kT

N

F kT e d d N





 

 

q q

q q

12 12

( )

R

U r

12

r

Interaction potential

• ignore attractive interactions, keep only hard core

2/28/2018 23 ( ) , ,

1

1 ln ... ... !

R U i j i j i j kT

N

F kT e d d N





 

 

q q

q q

12 12

( )

R

U r

12

r

1 ln !

N

F kT G N  

Free energy

• no. of states of N distinguishable particles
• adding one particle

2/28/2018 24 ( ) , ,

1...

R U i j i j i j kT N

N N

G e d d



 

 

q q

q q

( ) ( ) 1 , , ,N 1

1 1 1

( ) ...

R R U U i N i j i i j i j i kT kT N

N N N

G e e d d d

   

     

  

q q q q

q q q

( ) 1 ,N 1

1 1 1 1

( ... ) ( ) ...

R U i N i i kT N

N N N N N

G G P e d d d

  

    



 

q q

q q q q q

( ) 1 ,N 1

1 1

R U i N i i kT

N N N

G G e d

  

   

  



q q

q

Mean field free energy

• no. of states of N distinguishable particles
• or
• and
• where

is the average excluded volume fraction.

2/28/2018 25 ( ) 1 1,

1

R U i i i kT

N N

G e d



 

 



q q

q

( ) 1 1,

1

( ) 1

R U i i i kT

W e





  

q q

q

( ) 1 1,

1

(1 (1 ))

R U i i i kT

N N

G e d



 

   



q q

q

1 1

( [1 ( )] )N

N

G W d



 



q q

• The mean field free energy is

where and where is the average volume effectively occupied by one particle.

Mean Field Free Energy

2/28/2018 26

1 1

1 ln [ (1 ( )) ] !

N

F kT W d N   



q q

3 1 1 1

d d d   q r

1

( )

eff

N W v V  q

eff

v

Effectively occupied volume*

• for spheres in dilute limit
• for close packed spheres
• for any number density

27

1 4 2

eff exc

v v V   3 2 3 2 1 8 6

eff exc exc

v v V V      ( )

eff exc

v V   

2/28/2018

1 1 2 6   

Excluded volume fraction

28

( )

eff exc

v V   

1 2

1 ( , ) 1 2 2

( ) ( ) ( )(1 )

U kT

W e d   

 

 



q q

q q q

2/28/2018

and at high densities,

1 2

1 ( , ) 1 2 2

( ) ( )(1 )

U kT

W e d  

 

 



q q

q q q

where is to be determined.



• now
• gives

– nb. at low densities,

Example: Isotropic system of spherical particles:

2/28/2018 29

1 1

1 ln [ (1 ( )) ] !

N

F kT W dq N   



q 4

• W

v   ln( 4 )

• V

F NkT v N    (1 4 )

• kT

P v    

Van der Waals equation of state

(1 4 )

• P

kT v   

3 1 1 1 1 1

1 ln [ (1 ( , )) , ] !

i

N i i

F kT W d d N      



r r

subvolume; particles in volume

Density functional form of the free energy

• divide system into subvolumes
• ~ constant in each subvolume
• isothermal (not necessary)
• free energy is additive, so

2/28/2018 30

System

i

N

V  

th

i ( )  q

• since is slowly varying,
• and

Density functional form of the free energy

2/28/2018 31

1 1 1

( , ) W  r

1 1

( , ) 1 1 1 1 1

1 ln [ {(1 ( , ))}] ( , )

V i

F kT W





 

   

r

r r



 

( , , , ) 1 1 2 2 12

3 2 1 1 1 1 1 1 3 2 3 2 1 1 2 2 2 2 1 1

( , )ln ( , ) ( , )ln(1 ( , )(1 ) )

R U kT

F kT d d kT e d d d d     



   

  

r r

r r r r r r r

 

      

Orientational Probability Distribution Function

• assume constant number density

– write where here is the number density, and is the PDF.

• then the free energy density is
• at low densities, we can expand, and get Onsager

2/28/2018 32

( , ) ( ) f      r

( ) 1 f d   



2 2 2 1 1 1 1 2 1 2 2 1

ln ( )ln ( ) ( )ln(1 ( ) ( , ) )

exc

f f d f f V d d kT                 

  

F

 ( ) f 

2 2 2 1 1 1 2 1 2 2 1 1

1 ( )ln ( ) ( )( ) ( , ) ) 2

exc

f f d f V d d kT             

 

F

Orientation descriptor

2/28/2018 33

• cylindrically symmetric particles
• unit vector along symmetry axis
• assume uniaxial, with average direction
• rientation descriptor
• excluded volume

ˆ l

ˆ l

ˆ n ˆ n

ˆ ˆ x   l n

12

 2a 2b

L W

1 2 2 1 2 2

( , ) ( ) ( )

exc

V x x C DP x P x  

Free energy to minimize

• dimensionless free energy density
• where and
• need to find to minimize .
• but before doing that……..

2/28/2018 34 1 1 1 1 1 1 1 2 2 2 2 1 2 1

ln ( )ln ( ) ( )ln(1 ( ( ) ( ) ( )d ) f x f x dx f x c d f x P x P x x dx kT        

  

F

( ) f x F c C   d D  

2/28/2018 35

diversion

Free energy with expanded log term

• dimensionless free energy density

– minimizing wrt gives – where .

2/28/2018 36

1

( ) f x  F

1 1 1 1 1 1 1 2 2 2 2 1 2 1

ln ( )ln ( ) ( ) ( ( ) ( ) ( )d ) f x f x dx f x c d f x P x P x x dx kT       

  

F

1 2

( ) ( ) S f x P x dx  

2 2

( ) ( )

( )

DSP x DSP x

e f x e dx

 





Modified Onsager theory

2/28/2018 37

Isotropic Nematic

first order unstable metastable stable nematic Isotropic

Order parameter S vs.

0.43

NI

S 



2 2

1 ( ) 2 1 ( )

( )

DP x DSP x

P x e dx S e dx

 

 



Equation of state

• free energy
• pressure
• and

2/28/2018 38

2 ( )

1 2

1 1 ln ln 2 2

DSP x

C DS e dx kT



         F P       F F +

2

1 (1 ( )) 2 P kT C DS     

Summary of diversion

• if term is expanded, have modified Onsager model
• behavior is equivalent to Maier-Saupe theory
• 1st order transition at SNI=0.43
• density plays role of inverse temperature
• system is not ‘hard’, density can be increased arbitrarily

2/28/2018 39

ln

2/28/2018 40

the ‘hard’ problem

Euler-Lagrange Equation

• dimensionless free energy density

– minimizing wrt gives – where is the order parameter.

2/28/2018 41

1

( ) f x  F

1 1 1 1 1 1 1 2 2 2 2 1 2 1

( )ln ( ) ( )ln(1 ( ( ) ( ) ( )d ) f x f x dx f x c d f x P x P x x dx     

  

F

1 2

( ) ( ) S f x P x dx  

2 2 2 1 2 2 2 2 2 2 2 1 2 2 2 2

( ) ( ) ( )1 ( ( )) 2 1 ( ) ( ) ( )1 ( ( )) 2 1 1 1

(1 ( ( )) (1 ( ( )) ( )

dP x P x f x dx c dSP x dP x P x f x dx c dSP x

f x c dSP x e c dSP d x e x

   

 

   

      



2 1

(1 ( ) f ( ) i c dSP x    

1

( ) 0 otherwise. f x 

Solution

2/28/2018 42

• define
• primary order parameter
• auxiliary order parameter
• PDF

1 c d     

12 2 12

( ) (cos )

exc

V c dP     

1 2

( ) ( ) S f x P x dx  

1 2 2

( ) ( ) 1 ( ) P x f x dx S P x    



~density; embodies shape information

2 2

( ) 2 ( ) 2

(1 ( )) ( ) (1 ( ))

P x P x

S P x e f x S P x e dx

    

 

 

  



2

( ( ) if ) SP x    ( ) 0 otherwise. f x  1 , 0; 1, c d          

Order parameter & phase behavior

2/28/2018 43

nematic isotropic

I N

S 

0.224

NI

   0.545

NI

S 

vs S 

   

Primary and auxiliary order parameters

2/28/2018 44

1 2

( ) ( ) S f x P x dx  

I N

S  

1 2 2

( ) ( ) 1 ( ) P x f x dx S P x    



 

• physical significance of is not well understood

– ‘effective field’: gives high degree of order near critical density

Primary and auxiliary order parameters

2/28/2018 45

  S

1 2 2

( ) ( ) 1 ( ) P x f x dx S P x    



1 2

( ) ( ) S f x P x dx  

vs S 

Cutoff

2/28/2018 46

• integration limits indicate allowed orientations

– solve self-consistently – and

• obtain lower limit of integration

( ) f x 

2 2

( ) 1 2 2 2 ( ) 1 2

( ) 1 2 2 3 3 3 ( )

P x x P x x

P x e x P x e dx dx

   

 

 

   



1 2

( ) ( )

x

S f x P x dx  

1 2 2

( ) ( ) 1 ( )

x

P x f x dx S P x    



x

Cutoff: lower limit

2/28/2018 47

I N

S   xo

Cutoff

2/28/2018 48

• integration limits indicate allowed orientation

– solve self-consistently – and

• obtain upper limit of integration

( ) f x 

2 2

( ) 2 2 2 ( ) 2

( ) 1 2 2 3 3 3 ( )

P x P x x x

P x e x P x e dx dx

   

 

 

   



2

( ) ( )

x

S f x P x dx  

2 2

( ) ( ) 1 ( )

x

P x f x dx S P x    



x

Cutoff: upper limit

2/28/2018 49

I N

S   xo

Cutoff

• physical reason for cutoff?
• mean field model, all particles obey same PDF.
• ‘hard’ constraint on density
• if the PDF allows a specific orientation, then all particles

may adopt that orientation, and the density constraint would be violated

• mean field + density constraint require the cutoff

2/28/2018 50

Orientational distribution f(x)

51

S 

2/28/2018

1 c d     

   

Orientational distribution f(x)

52

S

2/28/2018

 1 c d     

   

Orientational distribution f(x)

53

S

2/28/2018

 1 c d     

   

Orientational distribution f(x)

54

S

2/28/2018

 1 c d     

   

Special point: S undetermined

• consider
• if then and
• can take on any value at ;

– positivity of PFD, requires that .

2/28/2018 55

 

xo 

2 2

( ) 1 2 2 ( ) 1 2

( )(1 ( )) (1 ( ))

P x P x

S P x P x e S S P x e dx dx

   

 

 

  

 

2 2

( ) 1 2 2 2 ( ) 1 2

( ) 1 2 2 3 3 3 ( )

P x x P x x

P x e x P x e dx dx

   

 

 

   



2 2

( ) 1 2 ( ) 1 2

( ) (1 ( ))

P x P x

P x e dx S P x e dx

   

 

 

 

 

S

2 2

1 ( ) 5 P x      

2( )

SP x    0.2 0.4 S   

Special point: S undetermined

• consider
• if then and
• can take on any value at ;

– positivity of PFD, requires that .

2/28/2018 56

 

1 xo 

2 2

( ) 1 2 2 ( ) 1 2

( )(1 ( )) (1 ( ))

P x P x

S P x P x e S S P x e dx dx

   

 

 

  

 

2 2

( ) 2 2 2 ( ) 2

( ) 1 2 2 3 3 3 ( )

P x P x x x

P x e x P x e dx dx

   

 

 

   



2 2

( ) 1 2 ( ) 1 2

( ) (1 ( ))

P x P x

P x e dx S P x e dx

   

 

 

 

 

S

2 2

1 ( ) 5 P x      

2( )

SP x    0.2 0.4 S   

Special point

2/28/2018 57

nematic isotropic

I N

S  free energy is independent of S

   

Non-admissible order parameter region

• free energy with non-equilibrium order parameter

– minimizing wrt gives – where is determined by .

2/28/2018 58

1

( ) f x  F

1 1 1 1 2 1 2

( )ln ( ) ( )ln( ' ( )( ( ) 1 ( ( ) ) ) ) f x f x dx f x c f x dSP x dx S P x dx        

  

F

2 2

' ( ) 2 ( ) 2 '

(1 ( )) ( ( ) 1 ( ))

P x P x

S P x e S f x dx P x e

 

    



2

( ( ) if ) SP x   

1

( ) 0 otherwise. f x  ' 

1 2

( ) ( ) S f x P x dx  

• for and
• cutoff occurs when
• but
• no solution when

2/28/2018 59

Non-admissible order parameter region

2 2

' ' 1 ( ) 2 2 1 ( ) 2

( ( )(1 ( )) (1 ( ))

P x x P x x

S P x P x e S P d S d x x x e

 

    

 

2(

) 1 SP x  

2(

) S P x 

2

S   as is decreased ' S      S  ( ~1) x

Non-admissible order parameter region

2/28/2018 60

non-admissible region

I N

S 

   

• for and
• cutoff occurs when
• but
• no solution when

2/28/2018 61

Non-admissible order parameter region

2 2

' ' ( ) 2 2 ( ) 2

( ( )(1 ( )) (1 ( ))

x P x x P x

S P x P dx S d x e S P x x e

 

    

 

2(

) 1 SP x  

2(

) S P x 

2

S   as is increased ' S      S  ( ~ 0) x

Non-admissible order parameter region

2/28/2018 62

non-admissible region no solutions exist for any S

I N

S 

   

Solutions and energy landscape

• free energy as function of non-equilibrium

2/28/2018 63

• – where is determined by .
• plot vs.

2 2

' ( ) 2 ( ) 2 '

(1 ( )) ( ( ) 1 ( ))

P x P x

S P x e S f x dx P x e

 

    



2

( ( ) if ) SP x    ( ) 0 otherwise. f x 

1 2

( ) ( ) S f x P x dx  

1 1 1 1 1 2 2

( ( )ln ( ) ( )ln( )( ( ) )) ( ) ) f x f x dx f x SP f x S P x dx x dx       

  

F F

S ' 

F

S

Solutions and energy landscape

2/28/2018 64

I N

    

Solutions and energy landscape

2/28/2018 65

I N

0.5  

   

vs S F

Solutions and energy landscape

2/28/2018 66

I N

0.05  

   

vs S F

Solutions and energy landscape

2/28/2018 67

I N

0.1   

   

vs S F

Solutions and energy landscape

2/28/2018 68

I N

0.15   

   

vs S F

Solutions and energy landscape

2/28/2018 69

I N

0.2   

   

vs S F

Solutions and energy landscape

2/28/2018 70

I N

0.231   

   

vs S F

Equation of State

• equation of state can be obtained from the free energy

– the pressure is – and explicitly

• density cannot be increased without limit!

2/28/2018 71 2

ln ( )ln ( ) ( )ln(1 ( ( )) kT kT f x f x dx kT f x c dSP x dx          

 

F P        F F

2

1 ( )1 ( ( )) P kT f x dx c dSP x     



2

1 ( ) kT P c dS     

Nematic-isotropic coexistence

2/28/2018 72

 

double tangent denotes equal pressures horizontal tie line denotes equal chemical potentials

Comparison of theory and simulations

– fit:

2/28/2018 73

• v



1 6  

Volume fraction at transition

• vs. inverse eccentricity

P.J. Camp, C.P. Mason, M.P. Allen, A.A. Khare and D.A. Kofke,

• J. Chem. Phys. 105, 2837 (1996)
• D. Frenkel and B.M. Mulder,
• Mol. Phys. 55, 1171 (1985)
• G. Odriozola, J. Chem. Phys.

136, 134505, (2012) prolate and oblate hard ellipsoids

Summary

2/28/2018 74

Jamie Taylor

• solutions via dual optimization scheme
• showed

– existence of global minimizer in terms of – all minimizers other than isotropic have cutoff region – regularity of PDF

• some numerics
• all JT results are fully consistent with ours

2/28/2018 75



Summary

2/28/2018 76

• provided ‘hard’ free energy for ellipsoidal particles
• obtained minimizers as function of
• examined existence and stability of solutions
• 1st order NI transition at &
• rich bifurcation landscape; N-I coexistence
• cutoff in PDFs for all non-isotropic solutions; mean field
• good agreement of model with existing MC simulations.

 0.224

NI

   0.545

NI

S 

Summary

2/28/2018 77

S

 vs. S 

• E. Nascimento, P.P-M., J.Taylor, E.Virga and X, Zheng, Phys. Rev. E 96, 022704 (2017)

Outstanding questions

• significance of auxiliary order parameter
• possibility of biaxiality
• agreement of pressure dependence of with

experiment

• effects of fields
• phase separation behavior

2/28/2018 78



S , E H

Future work

• include attractive interaction

– London dispersion

in model, and

• compare predictions

– pressure dependence of S – isotropic – nematic coexistence

with experiment.

2/28/2018 79