Statistically kinked chain

More realistic is the model of the statistically kinked chain with valence angles and hindrances to internal rotation. Here the mean square length is derived as ... 

We must however not forget that the macromolecules of gum arabic and gelatin are set along the molecule chain with numerous different groups which are polar (contain dipoles). For these kinds of molecules the considerations on the randomly kinked molecule in Chapter IV certainly do not hold rigidly, unless perhaps in extremely dilute sols. It is probable that the molecular skeins of these biocolloids differ from the theoretical ideal of Chapter IV by the additional statistical occur-... 

The principles of the modem physical theories of the deformation of rubber xerogels have already been dealt with in Chapter IV, p. 123. If a strip of raw rubber is rapidly extended, the molecules, which initially assumed randomly kinked forms, are stretched too. The more extended shape of the chains is a statistically less probable one corresponding to a lower entropy. en the piece of rubber is rapidly released again, it reassumes its original form in that the chains return to their, most probable configurations The entropy-character of rubber elasticity has been proven in that it exhibits a positive temperature coefficient. 

The chains extending themselves between the primary crystalline junction points (cf. p. 497) will, just as in rubber, possess a randomly kinked statistically most probable configuration and may be responsible for the elastic behaviour. 

Since hydrocarbon chains in the Uquid state are never fully extended, an effective chain length, /etf, can be defined that gives the statistically most likely extension as calculated by the same procedure used for the calculation of polymer chain dimensions. For a chain with n c = 11, the ratio of /max to /ett will be approximately 0.75. In the micellar core, because of restrictions imposed by the attachment of the hydrocarbon tail to the head group bound at the surface, the mobility of the chains may be significantly limited relative to that of bulk hydrocarbon chains. The presence of kinks or gauche chain conformations, which may be imposed by packing considerations, will result in a calculated Imax amounting to only about 80% of the theoretical maximum. 

An average way of taking into account those constraints is to assume that the chain is trapped into a virtual tube, envelope of all the obstacles which directly surround it (dotted line of (Fig. 2c)), a picture first proposed by Edwards. The chain can move freely along the curvilinear axis of the tube, but it cannot escape laterally. At any time, by fluctuations of its local kinks, the chain leaves some parts of the initial tube, and the chain extremities define new portions of the tube. The detailed statistical description of de Gennes leads to definite predictions for the molecular weight dependences of both the relaxation times and the diffusion coefficient of the chain, which we shall present in a simplified version. 

---