Kuhn segment

If the persistence length Ip is much larger than the mean chain diameter, d, Yamakawa and Fujii gave limiting values for ai = - ln(d/2Ip) and = 0.1382. Freire and Garcia de la Torre [122] have considered further these coefficients. The factor 2Ip appears rather than Ip simply because 2Ip is equivalent to the statistical Kuhn segment length... 

Fig.1. Regular star macromolecules with/=3,4, and 8 arms of identical length. The arms or rays can consist of rather stiff chains, but are in most cases flexible chains. The global structure is determined by the overall shape of the whole macromolecule the internal structure is indicated by a domain that is much smaller than the overall dimension but still larger than a few Kuhn segments...

We begin by formulating the free energy of liquid-crystalline polymer solutions using the wormlike hard spherocylinder model, a cylinder with hemispheres at both ends. This model allows the intermolecular excluded volume to be expressed more simply than a hard cylinder. It is characterized by the length of the cylinder part Lc( 3 L - d), the Kuhn segment number N, and the hard-core diameter d. We assume that the interaction potential between them is given by... 

Fig. 14. The Kuhn segment number N dependence of the axial ratio Le/d for fuzzy cylinders with different d [104]...

Values of B calculated from the ordinate intercepts are shown in Fig. 23 as a plot of B/(2q)3 against the number of the Kuhn segments N. For N<4, the data points for the indicated systems almost fall on the solid curve which is calculated by Eq. (78) along with Eqs. (43), (51), (52), and Cr = 0. A few points around N 1 slightly deviate downward from the curve. Marked deviations of data points from the dotted lines for the thin rod limit, obtained from Eq. (78) with Le = L and de = 0, are due to chain flexibility the effect is appreciable even at N as small as 0.5. The good lit of the solid curve to the data points (at N 4) proves that the effect of chain flexibility on r 0 has been properly taken into account by the fuzzy cylinder model. 

Table 1. Molecular-weight averages, monomer lengths (/) and estimated upper value of the Kuhn segment lengths (/k) of the poly[2]catenanes 48, 49, and 59, and the poly[2]catenands 51, 56.

R(. = r(. r. j is the end-to-end vector of Kuhn segment i and A.. is the Rouse matrix that couples the beads together through the spring force interactions. The spring force coefficient for a single spring in the sequence will be simply NK times the... 

If we return to the chain, z is the number of Kuhn segments in the chain, and l is the length of the segment. To avoid uncertainty, one adds a condition which is usually zl = M, so that one has a definition of the length... 

Formulae (1.2) and (1.3) determine the model of a freely-jointed segment chain, which is frequently used in polymer physics as a microscopic heuristic model (Mazars 1996, 1998, 1999). A Kuhn segment in the flexible polymers (polyethylene, polystyrene, for example) usually includes a few monomer units, so that a typical length of the Kuhn segment is about 10 A or 10-7 cm and, at the number of segments 2 = 104, the end-to-end distance (R2)1 2 of a macromolecule is about 10-5 cm. 

The temperature dependence of the size of a macromolecular coil is included in the coefficient of stiffness C T) which has the meaning of the ratio of the squared length of a Kuhn segment to the squared length of the chemical bond, and can be calculated from the local chemical architecture of the chain. The results of the calculations were summarised by Birshtein and Ptitsyn (1966) and by Flory (1969). 

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