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Tilting Transition in a Liquid Crystalline Polymer Brush Steven - - PowerPoint PPT Presentation
Tilting Transition in a Liquid Crystalline Polymer Brush Steven - - PowerPoint PPT Presentation
Tilting Transition in a Liquid Crystalline Polymer Brush Steven Blaber , Nasser Abukhdeir and Mark Matsen University of Waterloo Motivation Previous Work: Amoskov, Birshtein, and Pryamitsyn; Macromolecules 1996 Limitations Freely jointed
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Previous Work: Amoskov, Birshtein, and Pryamitsyn; Macromolecules 1996
Limitations
◮ Freely jointed chain model (no bending penalty) ◮ Director fixed along z (no tilting)
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Theory
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Wormlike Chain Model
Fixed contour length controlled by the degree of polymerization N and segment length b ℓc = bN . The penalty for bending in the worm-like chain model is UB = κ 2N 1 ds
- du(s)
ds
- 2
, with κ a dimensionless bending modulus, which controls the persistence length, ℓp = bκ .
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Implicit Solvent
We will be using an implicit solvent model, which assumes a semi-dilute mixture. This corresponds to expanding the solvent entropy Ss = −
- dzφs ln φs ,
with φs = 1 − φ , assuming low polymer concentration φ ≪ 1 Ss =
- dz
- −φ + φ2
2 + φ3 6 + O(φ4)
- The linear term does not effect the system and the quadratic term is
combined with the Flory-Huggins polymer-solvent interaction parameter, χ, into one parameter Λ0 with energy U0 = Λ0 2
- dzφ2(z) .
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LC Interactions
Maier-Saupe interactions: U2 = −Λ2 2
- dzdudu′φ(z, u)P2(u · u′)φ(z, u′) ,
where φ(z, u) is the concentration with orientation u and P2 is the second Legendre polynomial. Tensor order parameter: Qij(z) ≡ 3 2φ(z)
- du
- uiuj − δij
3
- φ(z, u) ,
i & j = x, y, z = 3S(z) 2
- ni(z)nj(z) − 1
3δij
- + P(z)
2 (li(z)lj(z) − mi(z)mj(z)) , where S is the uniaxial and P the biaxial scalar nematic order parameters while n, l, m are orthogonal unit directors.
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Self-Consistent Field Theory (SCFT)
Replace interactions on a polymer by a field, w(z, u), Ufield =
- dzduw(z, u)φ(z, u) .
◮ Calculate φ(z, u) for a given w(z, u) ◮ Adjust w(z, u) to satisfy self-consistent equation w(z, u) = Λ0φ(z) − Λ2
- du′P2(u · u′)φ(z, u′) .
This process yields the free energy F = − ln Q − 1 2
- dzduw(z, u)φ(z, u)
◮ Initially we assume alignment along z axis = ⇒ azimuthal symmetry, so w(z, u) → w0(z, uz) .
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Results
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Concentration and Scalar Order Parameter
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Backfolded State: Director Fixed Along z axis
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Backfolded State: Director Fixed Along z axis
Utot = 1 2
- dz
- Λ0 − Λ2S2(z)
- φ2(z) + UB
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Tilted State: Director Free
Utot = 1 2
- dz
- Λ0 − Λ2S2(z)
- φ2(z) + UB
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Tilted State: Director Free
Field equation β[w] ≡ w(z, u) − Λ0φ(z) + Λ2
- du′P2(u · u′)φ(z, u′) .
Jacobian Jz,z′′,u,u′′ ≡ Dβ[w] Dw(z′′, u′′)
- w0
. The solution is no longer stable when an eigenvalue of the Jacobian becomes negative. The corresponding eigenvector δw(z, u) represents the change in the field that will reduce the free energy.
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Tilted State: Director Free
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Tilted State: Director Free
Utot = 1 2
- dz
- Λ0 − Λ2S2(z)
- φ2(z)
- Uint(z)
+UB
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Tilted State: Director Free
Utot = 1 2
- dz
- Λ0 − Λ2S2(z)
- φ2(z) + UB
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Phase Diagram
Utot = 1 2
- dz
- Λ0 − Λ2S2(z)
- φ2(z) + UB
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Phase Diagram
Utot = 1 2
- dz
- Λ0 − Λ2S2(z)
- φ2(z) + UB
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Phase Diagram
Utot = 1 2
- dz
- Λ0 − Λ2S2(z)
- φ2(z) + UB
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Tilt Angle
Utot = 1 2
- dz
- Λ0 − Λ2S2(z)
- φ2(z)
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Tilt Angle
Ss =
- dz
- −φ + φ2
2 + φ3 6 + O(φ4)
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Tilt Angle
Utot =
- dz
1 2
- Λ0 − Λ2S2(z)
- φ2(z) + C
3 φ3(z)
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Conclusion
◮ Strong LC interactions can nematically collapse a polymer brush; however, it does so by tilting rather than backfolding. ◮ The transition is an instability within the implicit solvent model and is continuous once you add higher order corrections.
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