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 chain model (no bending penalty) ◮ Director fixed along z (no tilting)
Theory
Wormlike Chain Model Fixed contour length controlled by the degree of polymerization N and segment length b ℓ c = bN . The penalty for bending in the worm-like chain model is � 1 � � 2 d u ( s ) U B = κ � � ds , � � 2 N � ds � 0 with κ a dimensionless bending modulus, which controls the persistence length, ℓ p = b κ .
Implicit Solvent We will be using an implicit solvent model, which assumes a semi-dilute mixture. This corresponds to expanding the solvent entropy � S s = − dz φ s ln φ s , with φ s = 1 − φ , assuming low polymer concentration φ ≪ 1 � � � − φ + φ 2 2 + φ 3 6 + O ( φ 4 ) S s = dz 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 Λ 0 with energy � U 0 = Λ 0 dz φ 2 ( z ) . 2
LC Interactions Maier-Saupe interactions: � U 2 = − Λ 2 dzd u d u ′ φ ( z , u ) P 2 ( u · u ′ ) φ ( z , u ′ ) , 2 where φ ( z , u ) is the concentration with orientation u and P 2 is the second Legendre polynomial. Tensor order parameter: � � � 3 u i u j − δ ij Q ij ( z ) ≡ d u φ ( z , u ) , i & j = x , y , z 2 φ ( z ) 3 � � = 3 S ( z ) n i ( z ) n j ( z ) − 1 + P ( z ) 3 δ ij ( l i ( z ) l j ( z ) − m i ( z ) m j ( z )) , 2 2 where S is the uniaxial and P the biaxial scalar nematic order parameters while n , l , m are orthogonal unit directors.
Self-Consistent Field Theory (SCFT) Replace interactions on a polymer by a field, w ( z , u ), � U field = dzd u w ( z , u ) φ ( z , u ) . ◮ Calculate φ ( z , u ) for a given w ( z , u ) ◮ Adjust w ( z , u ) to satisfy self-consistent equation � d u ′ P 2 ( u · u ′ ) φ ( z , u ′ ) . w ( z , u ) = Λ 0 φ ( z ) − Λ 2 This process yields the free energy � F = − ln Q − 1 dzd u w ( z , u ) φ ( z , u ) 2 ◮ Initially we assume alignment along z axis = ⇒ azimuthal symmetry, so w ( z , u ) → w 0 ( z , u z ) .
Results
Concentration and Scalar Order Parameter
Backfolded State: Director Fixed Along z axis
Backfolded State: Director Fixed Along z axis � U tot = 1 � � Λ 0 − Λ 2 S 2 ( z ) φ 2 ( z ) + U B dz 2
Tilted State: Director Free � U tot = 1 � � Λ 0 − Λ 2 S 2 ( z ) φ 2 ( z ) + U B dz 2
Tilted State: Director Free Field equation � d u ′ P 2 ( u · u ′ ) φ ( z , u ′ ) . β [ w ] ≡ w ( z , u ) − Λ 0 φ ( z ) + Λ 2 Jacobian � D β [ w ] � J z , z ′′ , u , u ′′ ≡ . � D w ( z ′′ , u ′′ ) � w 0 The solution is no longer stable when an eigenvalue of the Jacobian becomes negative. The corresponding eigenvector δ w ( z , u ) represents the change in the field that will reduce the free energy.
Tilted State: Director Free
Tilted State: Director Free � U tot = 1 � � Λ 0 − Λ 2 S 2 ( z ) φ 2 ( z ) + U B dz 2 � �� � U int ( z )
Tilted State: Director Free � U tot = 1 � � Λ 0 − Λ 2 S 2 ( z ) φ 2 ( z ) + U B dz 2
Phase Diagram � U tot = 1 � � Λ 0 − Λ 2 S 2 ( z ) φ 2 ( z ) + U B dz 2
Phase Diagram � U tot = 1 � � Λ 0 − Λ 2 S 2 ( z ) φ 2 ( z ) + U B dz 2
Phase Diagram � U tot = 1 � � Λ 0 − Λ 2 S 2 ( z ) φ 2 ( z ) + U B dz 2
Tilt Angle � U tot = 1 � � Λ 0 − Λ 2 S 2 ( z ) φ 2 ( z ) dz 2
Tilt Angle � � � − φ + φ 2 2 + φ 3 6 + O ( φ 4 ) S s = dz
Tilt Angle � � 1 � φ 2 ( z ) + C � � Λ 0 − Λ 2 S 2 ( z ) 3 φ 3 ( z ) U tot = dz 2
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.
Future Work ◮ Use an explicit solvent model to determine the tilt angle as a function of Λ 2 . ◮ A strongly stretched LCP brush, swollen with either a melt of LCs or LCPs could induce local orientational order in the melt at the interface which has potential applications for LC displays.
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