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

3.1. INTRODUCTION 55where the tensor order parameter is given by: q ij = q ⊥ δ ij + (q ‖ − q ⊥ )n i n j .If θ is the angle that a monomer makes with the nematic direction, n, thenq ‖ = 〈cos 2 θ〉 and q ⊥ = 1 2 〈sin2 θ〉. Since v is perpendicular to u it has theaverage〈v α i vβ j 〉 vu = 1 2 δ αβ(δ ij −q ij ) = 1 2 δ αβM ij (3.7)These averages can be used to evaluate the joint probability distribution byaveraging over delta functions used to count the configurations. Exponentiatingthese delta functions and expanding down from the resulting exponentallows their evaluation. The result is〈 (W(R,V) = δ R− ∑ ) ((au α +bv α ) δ V− ∑ )〉u α ×v ααα×∫ ∫ dηdζ=(2π) 6exp(iη·R+iζ ·V)〈 ( ) 〉∑1−i η ·(au α +bv α )+ζ ·(u α ×v α ) − 1 2 (···)2 +··· .αThe first few averages of this expansion can be evaluated, ignoring termsof order ( b 2,a)and then re-exponentiate and using the method of steepestdescents to evaluate the integral. An iterative method can be used to solvefor the points of steepest descent. The most interesting term comes from thecubic part. The resulting distribution function is( −3W (R,V) ∝ exp2La RT ·l −1 ·R− 1 )N VT ·M −1 ·V(× 1+ 3b [ ])aL 2 R×l −1 ·R ·M −1 ·V+O(R 4 ,R 2 V 2 ) ,where l = δ + (r −1)nn ≈ 3q. This tensor provides information on theanisotropy of the chain shape. The anisotropy of the chains is given by r =l ‖ /l ⊥ . Theimportant correction term is the couplingbetween RandV. Usingthis probability distribution the average binormal, V, can be calculated.∫〈V〉 ∝dVexp= 3bN2aL 2R×l−1 ·R,(− 1 )N VT ·M −1 ·VV(1+ 3b [ ] )aL 2 R×l −1 ·R ·M −1 ·Vwhereclearly in the ∫ dV the first(linear) term gives zero, so theO(V 2 ) termsmust be evaluated. In a rubber the polymer chains will have their ends fixed.It is assumed that deformations move these ends affinely. If the quenched-inend-to-end vector is denoted by R 0 then after a deformation, λ, the end-toenddistance becomes R = λ · R 0 . The quenched-in binormal vector, after