Statistical models of elasticity in main chain and smectic liquid ...

2.A. GEOMETRY OF A HAIRPIN AT ZERO TEMPERATURE 372.A Geometry of a hairpin at zero temperatureA polymer chain with a bend constant, B, in a nematic field with strength Jhas a free energy of the form∫ [L( )1 dθ 2F = F 0 + ds2 B + θ]1ds 2 J sin2 . (2.103)0The hairpin configuration which has the minimum free energy and obeys thefollowing boundary conditions is requiredθ = 0 ; s → −∞θ = π ; s → ∞Minimising the free energy expression Eq. (2.103) with respect to θ results inthe Euler-Lagrange equation:d 2 θds 2 = 12lh2 sin2θ, (2.104)where l h = √ (B/J). This equation has the solution:θ = 2tan −1[ ]e s/l h. (2.105)This shows that the hairpin has a size of order l h . The misalignment diesaway exponentially from the hairpin thereafter. As a result hairpins have aseparable character. Thesize becomes very large if the nematic field gets weakor if the bending energy cost becomes very high. Substituting this expressionback into the expression for the energy of the hairpin gives the result:u h = 2(BJ) 1/2 . (2.106)