Liquid Crystalline Ordering of Wormlike Polymers: Bulk and Confined Systems

1. Liquid Crystalline Ordering of Wormlike Polymers: Bulk and Confined Systems with Dominik Düchs Xiangqun Yuan Raul Cruz Hidalgo Jeff Z.Y. Chen

2. Outline • Introduction • Polymers and liquid crystal phases • Wormlike-chain model • Ideal chains • Semenov-Khokhlov (excluded volume) model for interactions (isotropic-nematic transitions) • Extension of S-K to include smectic (“end-segment” interactions) • Self-consistent field theory • Results • Nematic (and isotropic) – smectic transitions: dependence on elongation and flexibility • Isotropic-nematic transition of a fluid confined between two parallel hard walls

3. Polymers polypropylene “Stick-bead” model “Wormlike chain” model

4. LiquidCrystalPhases • Isotropic • Nematic • Smectic-A director

5. Questions • How to account for the smectic-A phase of semiflexible polymers: Recent experiments: Dogic and Fraden, PRL 78, (1997): suspensions of fd virus Li and de Jeu, PRL 92, (2004): polypropylene melt Muresan et al., Eur. Phys. J. E. 19 (2006) “main-chain” LC polymers • Confinement effects (on I-N transition): • Relevant to LC displays. • Confined DNA (in cells and viruses) (?) • Previously studied using “rigid-rod” models.

6. persistence length: Wormlike Chains bending energy The statistical probability of an ideal wormlike chain is

7. Interactions • excluded volume interactions (Onsager 2nd-virial approximation: Khokhlov and Semenov, 1981-82). (suitable for lyotropic liquid crystals) Can touch but not penetrate

8. Interaction Energy (2-body) ti = si/L N = number of polymers “sums” over all segments(t1, t2) of chains i and j sum over all pairs (i,j) of chains Can express as where (Microscopic Density)

9. Replace by a “local” contact potential (typical for polymers) where

10. Semenov-Khokhlov model for of homogeneous chains: (based on Onsager 2nd-virial approx). Neglect t -dependence of V. (all segments are equivalent) L D (Minimized when u1 and u2 are parallel) Derived by Semenov and Khokhlov by breaking a chain of length L into a large number n of discrete rigid segments of length  = L/n < p. The total interaction between two chains is the sum over interactions between pairs of such segments. Passing to the continuum limit (n), (Factors of n absorbed into V: n = L)

11. where the “contour-averaged” density is However, this model neglects differences in interactions between “end segments” (or “ends” e) and “interior segments” (or “segments” s). For nematic-isotropic phase transitions, corrections due to these interactions are of order D/L. But for a high degree of orientational order, the SK segment-segment interaction reduces to a magnitude of the same order. Hence, we add the “end-segment” interactions. In the local approximation: For hemispherical end caps

12. Ves describes “repulsion” between (or “segregation” of) “interior segments” and “ends”.

13. Mean-field approximation(or self-consistent field theory, SCFT) • Replace microscopic density by averaged density ( and ): single-molecule probability distribution function, satisfying the normalization condition where integrates over all chain conformations:

14. Helmholtz “free-energy functional”: • Minimizing F(b) with respect to m under the • constraint (a) gives…..

15. “mean field” where and

16. In summary Solution of theory: depends on Can express in terms of conditional distributions or probagators which satisfy “diffusionlike” equations where

17. Phase diagram for rigid molecules: ( = ) Theory: 1st order 2nd order Poniewierski

18. Effects of flexibility ( = 1)

19. Nematic-smectic transition line (vol. fraction  vs. inverse persistence length) for L/D = 6

20. Summary • Results suggest that smectic may become unstable for very flexible molecules ( < 1). • Need to resolve instabilities of calculations for L/D > 10. • Need to account for high-density corrections (e.g., “Parsons-Lee” approach).

21. Polymer confinement Recent studies: hard rigid rods between two smooth hard walls: • van Roij, Dijkstra, Evans, J. Chem. Phys. 113, 7689 (2000). • Chrzanowska et al. , J. Phys.: Condens. Matter 13, 4715 (2001). • Harnau and Dietrich, Phys. Rev. E 66, 051702 (2002). • Lagomarsino et al., J. Chem. Phys. 119, 3535 (2003). • de las Heras et al., J. Chem. Phys. 120, 4949 (2004).

22. Ordering of rod-like segments near a hard wall Favoured alignment of rods is parallel to wall. x Segments near wall have a restricted range of orientations z wall In-plane ordering (as seen from above) y z Uniaxial (random planar) Biaxial (2D nematic)

23. For a single hard wall (semi-infinite system) Chen and Cui (1995) showed that (in the “flexible limit” L >> P >> D) • Uniaxial–biaxial transition at wall occurs at chemical potential  (or density) about 17 % lower than bulk I-N transition at IN. • The nematic phase (with parallel alignment) completely wets the wall-isotropic interface as   IN.

24. Structure of confined wormlike chains Density and order parameter profiles: (a = “Kuhn length” = 2p ) Uniaxial: Sz = Sy = |Sx|/2 Biaxial: Sz > |Sx|/2

25. Phase diagram of confined wormlike chains W = wall separation a = “Kuhn length” = 2p