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 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 2/28/2018 4

Outline • motivation • system under consideration • free energy • the probability distribution function & order parameters • summary • outstanding questions 2/28/2018 5

motivation 2/28/2018 6

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 Gerardo Odriozola, J. Chem. Phys. 136 , 134505, (2012) 2/28/2018 9

Onsager Theory 2 V 1 N • free energy     F kT N [ ln( ) V ..] exc N 2 V • where w ( r )  12     3 kT V ( e 1) d r exc 12 is the pair excluded volume. 4    3 for hard spheres: V 8 r 8 v exc 0 0 3 Question: what is the region of validity? 2/28/2018 10

Onsager Theory: region of validity • key step in derivation: 2 2 N N – (ANYAS Eq. 19 – 21)    ln(1 V ..) V exc exc 2 V 2 V 2 N V  18 • but 10 !?! exc 2 V • Onsager makes no comment on this – refers to work of Mayer Nv – suggests region of validity  0 0.2 V • . 2/28/2018 11

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. * 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) 2/28/2018 12

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 P - pressure V - volume • describe bulk behavior N - no. of particles k - Boltzmann's constant  T - temperature – ideal gas: PV NkT – Problem: for a given , can be arbitrarily large! V N  11 - stress G - shear modulus 1     – neo-Hookean elasticity: 2  G ( ) - stretch  11 – Problem: stretch can be arbitrarily large! 2/28/2018 14

Guidance: Equations of state P - pressure • describe bulk behavior V - volume N - no. of particles k - Boltzmann's constant  – ideal gas: PV NkT T - temperature – Problem: for a given , can be arbitrarily large! V N – Non-ideal: Van der Waals NkT    2 (P a)   (V ) Nv - stress o 11 G - shear modulus  - stretch 1     – neo-hookean elasticity: 2 G ( )        11 2 2 2 I 1 1 2 3 – Problem: stretch can be arbitrarily large!  1 I 3 – Non-ideal: Gent     2 m G ( )( )   11 I I 1 m 2/28/2018 15

Origins of the ‘hard’ response   1 1 v      • VdW free energy 2 o F a kT ln( )  2 v o  1 I 3     1 W G (I 3)ln(1 ) • Gent energy density  m 2 I 3 m kT   • Helmholtz free energy density F ln Z V   Z  – free energy as available phase space . F 0 - must keep logarithm for ‘hard’ response! 2/28/2018 16

system under consideration 2/28/2018 17

System under consideration • assembly of identical hard ellipsoids • length width . 2 a L L W  • cannot interpenetrate 12 W 2 b    • pair excluded volume ( ) (cos ) V C DP exc 12 2 12 • volume one particle makes unavailable to the center of the other   • L W / and are known functions of the aspect ratio D C 4 ( L W 4 ( L W       C v 4 ) D v 2 ) o o 3 W L 3 W L 2/28/2018 18

Excluded volume • ellipsoids 2 a L      V ( ) C DP (cos ) 12 exc 12 2 12 W 2 b 1      2 V ( ) C D (3cos 1) exc 12 12 2    • isotropic average: V ( ) C exc 12   • parallel configuration: V (0) C D exc 2/28/2018 19

the free energy 2/28/2018 20

Single component system   • Helmholtz free energy F kT ln Z   d d q p  H ( , ) q p Z e ... kT • partition function:    U ( ) q • configurational partition function: Z e d q .... kT (integrate out momenta)   ( ) 1 Uij q q i j    • pairwise interactions: F kT ln e dq ... dq ... kT N m N ! 2/28/2018 21

Interaction potential • ignore attractive interactions, keep only hard core R ( ) U r 12 12 0 r 12 R  1 U ( q q ) i j i j ,    i j ,   ln ... ... F kT e d q d q kT 1 N N ! 2/28/2018 22

Interaction potential • ignore attractive interactions, keep only hard core R ( ) U r 12 12 0 r 12 R  1 U ( q q ) i j i j ,    i j ,   ln ... ... F kT e d q d q kT 1 N N ! 1   ln F kT G N N ! 2/28/2018 23

Free energy • no. of states of N distinguishable particles   R  U ( q q ) i j i j ,  i j , G e d q 1 ... d q kT N N  N • adding one particle    R R    U ( q q ) U ( q q )  i j i N 1 i j , i  i j ,  i ,N 1 ( ) ... G e e d q d q d q kT kT   N 1 N 1 1 N   N R   U ( q q )  i i N 1    i ,N 1  G G P ( q ... q ) ( e d q ) d q ... d q kT   N 1 N 1 N N 1 1 N   N R   U ( q q )  i N 1 i   i ,N 1    G G e d q kT   N 1 N N 1  2/28/2018 24

Mean field free energy • no. of states of N distinguishable particles R  U ( q q ) 1 i i   1, i   N G e d q kT N 1  R  U ( q q ) i 1 i   1, i     • or N G (1 (1 e )) d q kT N 1     ) N G ( [1 W ( q )] d q • and N 1 1  R  U ( q q ) 1 i i  1, i    • where W ( ) 1 e q kT 1 is the average excluded volume fraction. 2/28/2018 25

Mean Field Free Energy • The mean field free energy is 1     N F kT ln [ (1 W ( q )) d q ] 1 1 N !   3 where d d d q r 1 1 1 N  and where is the average volume v ( ) W q v eff 1 eff V effectively occupied by one particle. 2/28/2018 26

Effectively occupied volume* • for spheres in dilute limit 1   v 4 v V eff 0 exc 2 • for close packed spheres 3 2 3 2 1    v v V V   eff 0 exc exc 8 6 • for any number density    v ( ) V eff exc 1 1    2 6 2/28/2018 27

Excluded volume fraction    v ( ) V eff exc 1   U ( q q , )      1 2 W ( q ) ( ) ( q )(1 e kT ) d q 1 2 2  and at high densities, 1   U ( q q , )     1 2 W ( q ) ( q )(1 e kT ) d q 1 0 2 2   where is to be determined. 0 2/28/2018 28

Example: Isotropic system of spherical particles: 1     N ln [ (1 ( )) ] F kT W q dq 1 1 ! N • now   4 W v o V    F NkT ln( 4 v ) o N • gives  Van der Waals kT  P   equation of state (1 4 ) v o    – nb. at low densities, P kT (1 4 v ) o 2/28/2018 29