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

2.2. MODEL OF HAIRPIN CHAINS 23f g = − ∂F g∂R = k BTRl 2 N(2.57)⇒ k g = k BTl 2 N . (2.58)For the hairpin chain the asymptotic limit is used. This will provide an overestimateof the spring constant for smaller values of fN.F hp ≈ −k B T R2 fN2N 2 l 2 (2.59)f hp ≈ k B T fNN 2 l2R (2.60)⇒ k hp = k B T fNl 2 N 2 (2.61)This shows that when fN ∼ 5 the Gaussian chains are much stiffer than thehairpin chains. The estimate of the spring constant of the undulating chain isgiven in §2.E. The result obtained isThe ratio of moduli of chains isf z ≈ 4u hk B T⇒ k u = 4u hJk B TLJδR (2.62)L(2.63)≈ 4u hNl 2 (2.64)k u : k g : k hp = 4u hBl 2 : k BTNl 2 : k BTfNN 2 l 2 (2.65)here the combination fN is retained since it is an estimate of the number ofhairpins per chain, n hp . Then u hk B T ≈ ln Nn hp, using the definition of f. In anyevent, one expects u h > k B T because hairpins are well-defined and relativelyinfrequent events. Thus the ratio of the spring constants becomesk u : k g : k hp = 4ln N n hp: 1 : n hpN(2.66)From these estimates it is clear that when the hairpins are removed from achain it becomes significantly harder to stretch. The chains in this state canonly have their end-to-end distance increased by a small amount because theirlongitudinal spatial extent is near to their arc length.2.2.5 Perpendicular directionIn the plane perpendicular to the nematic direction the hairpin chains havetwo sources that can give rise to an end-to-end distance: Gaussian distributed