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

2.3. NON-AFFINE DEFORMATION MODEL 25a)01010101b)01010101Figure 2.9: The figure illustrates the elongation of the networkof hairpin chains. a) shows an undeformed section andb) shows the same section after a small deformation. The centralstrand has taken up most of the deformation leaving thesurrounding two unchanged.inert. As the rubber is stretched the inert chains remain unchanged whilstthe hairpinned strands have their lengths changed in proportion to their currentlength so that this population takes up all the macroscopic strain, andthe sample as a whole accommodates the required macroscopic strain. Thosethat are at their maximum length are not able to stretch any more and sodo not contribute to the total length change of the rubber. Two measures ofthe deformation are thus required: the microscopic deformation, λ, and themacroscopic deformation, Λ. Chains cannot deform affinely with the bulk, Λ,since some do not extend at all and others must take up more than their shareof deformation. To describe a hairpin chain we adopt the following reducedunits (as in §2.2.3) for arc length L and end-to-end distance parallel to thenematic field, R, in terms of a persistence length, lz = R l; N = L l(2.69)The macroscopic deformation, Λ, can be related to the microscopic deformation,λ as follows. The fraction of chains with an initial end-to-end distance zis given by P(z), excluding the straight chains. The fraction of straight chainsis denoted by g(λ). The normalisation condition for P(z) is thus∫ N1 = g(1)+2 P(z)dz, (2.70)0where g(1) is the initial fraction of the chains that are straight, and the symmetryofP(z) hasbeenusedtohalve theinterval ofintegration. Asthesampleis stretched, chains from the hairpinned population with lengths initially inthe interval [ N λ,N] fall into the straight population so the increased fraction