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

26 CHAPTER 2. HAIRPIN CHAIN ELASTOMERSof straight chains, g(λ), is given by∫ Ng(λ) = g(1)+2 P(z)dz. (2.71)N/λThe macroscopic deformation can then be identified with the new mean spanR Λ with respect to the initial mean span, R 1 :R Λ = g(λ)N +[1−g(λ)]λ〈z〉 hp (2.72)∫ N/λ( ) P(z)ΛR 1 = g(λ)N +[1−g(λ)]2 λz dz. (2.73)1−g(λ)The length of an average chain is defined as the weighted average of thestraightened chain length and the length of the hairpinned chains. The latter,λ〈z〉 hp consists of the average 〈z〉 hp of the population initially in the intervalz = (0, N λ) that is still hairpinned after extension λ, taken to its currentaverage length by multiplication by λ. The normalisation of the part of theprobability distribution for the hairpinned chains has been written explicitlyfor clarity. The average initial length of the chains is defined as above butwith Λ = λ = 10∫ NR 1 = Ng(1)+2 zP(z)dz. (2.74)0In both Eq. (2.74) and Eq. (2.73) the first term on the right hand side is thelength taken up by the fully extended chains and the second term is the lengthtaken upbytheremainingchainswhichareat various degrees ofextension andhave varying numbers of hairpins in them. Eq. (2.73) determines Λ[P(z),λ]in terms of the internal microscopic deformation and the initial span distribution.This is a departure from the usual affine deformation approximation fornetworks because of the hard constraints met when the hairpins are absent orare eliminated. The macroscopic deformation, defined in Eq. (2.73), can berewritten as[{(g(1)+2 ∫ }NN/λ P(z)dz N +2λ ∫ ]N/λ0zP(z)dzΛ =Ng(1)+2 ∫ N0 zP(z)dz . (2.75)The asymptotics of Λ can be calculated as followsdΛdλ = 2 ∫ NλR 10zP(z)dz. (2.76)It is clear that the gradient goes to zero as λ → ∞. The value of Λ, fromEq. (2.75), is thenΛ(∞) = 2N ∫ N0 P(z)dz +g 1NNg 1 +2 ∫ N0 zP(z)dz