Eulers Chain Relation

Using Ihe Euler s chain relation and the reciprocal identity [Further information 2,2]... 

This is an example for Euler s chain relation. Euler s chain relationship can be derived more formally with Jacobian determinants, as shown in Example 1.23. Rewriting Eq. (1.19) without arguments results in... 

For Euler s chain relation, we need to show that — — j I — ]... 

The distribution of crystallite orientation can therefore be expressed as a function of Euler angles w(a, ft, y) defined for 0 < a < 2n, 0 < j3 < jt, and 0 < y < 2n. The question raised above can now be rephrased as follows (1) When the pole distributions /, , < ) are known for a finite number of poles j = 1, 2, 3,.. ., v, is it possible to derive w(a, /, y)l A related question is (2) When w(a, /J, y) is known, is it possible to calculate t(, < ) for any crystallographic plane (hkl) If the answers to these two questions are both affirmative, it follows that when the pole distributions for a finite number of poles are experimentally determined, the pole distribution for any other pole can be calculated. This last possibility is a useful one, since this implies that if, for example, the intensity of (001) reflection is too weak to allow direct experimental determination of the orientation distribution of polymer chain backbones, measurements of (100), (010), (110), etc., might allow the (001) orientation distribution to be derived indirectly. 

While the evaluation of pi can be simphfied using Euler relations for certain classes of monomer structures, these Euler relations [50] do not apply, for example, to the three polyolefin chains depicted in Fig. lb because they have short side chains. (Of the monomer structures in Fig. la, the Euler relations for pi are valid only for PEE and PHI.) Therefore, the geometrical index pi = /Mi = nPVsi is evaluated by directly enumerating all sets of three sequential bonds (nP ) that traverse a monomer of species i. [46] siunmarizes the details of these calculations and tabulates values of p,- for several monomer structures, so we pass now to a consideration of how thermodynamic properties depend on n and p, in the high molecular weight, incompressible system limit of the LCT. 

---