By Francesco Maria Pompeo Colonna
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Extra info for Abregé de la doctrine de Paracelse et de ses Archidoxes
When we consider a sequence of three points on the chain, 1, 2, and 3, with Ds13 ¼ Ds12 þ Ds23 , then the free enthalpy change connected with the orientation change along the segment, Ds12 , is not involved in the free enthalpy change of the following segment. This means that the thermal average of the orientation correlation can be written as f ðDs1s þ Ds23 Þ ¼ f ðDs12 Þf ðDs23 Þ: ð1:9Þ The only function that fulﬁls this relation is f ðDsÞ ¼ expðqDsÞ. The unknown parameter q can be determined evaluating the elastic energy of a chain segment with help of the principle of equipartition of energy in thermal equilibrium (Nelson, 2008).
Due to the deformation, the length of the small elastic rod is changed to ds0 ¼ ds þ dDs dsð1 þ uðsÞÞ. 33 Description of a bent polymer chain in the worm-like chain model and in the freely jointed chain model. ~ R denotes a vector pointing from the initial to the end point of the chain. j27 28 j 1 Molecular Units strain of the chain at the position s. For simplicity, we assume that the nondeformed polymer is a straight rod of length L. The change length Ð L of the contour ÐL DL of the deformed chain now can be expressed by DL ¼ 0 dDs ¼ 0 uðsÞds.
The L total twist DW is given by DW ¼ 0 tot vðsÞds. If all deformations are elastic, the energy dW el ðsÞ stored in the segment ds can be approximated by a quadratic dependence on the deformations ~ b, u, and v: 1 dW el ¼ kB TðA~ b 2 þBu2 þ Cv2 þ DuvÞds: 2 ð1:1Þ As the energy is a scalar, mixed terms including the vector of the bending deformation do not contribute to the energy. We scale the energy coefﬁcients in units of the thermal vibration energy per degree of freedom, ð1=2ÞkB T, with kB as the Boltzmann constant.