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Chain displacement length distribution

Here x,y, and z represent the coordinates of one end of the chain with respect to the other. The chain displacement length r = (x + y + and l/j6 is the most probable value of r. It is assumed in this model that the cross-linkages are introduced at random into the undeformed, isotropic polymer. The chains are thus free to assume random conformations. Hence Eq. (8.33) also represents the distribution of chain coordinates before stretching.(6,7) After stretching by a factor a, along the z-axis the distribution of chain coordinates becomes... [Pg.374]

Fig. 77.—Gaussian density distribution of the chain displacement vec tors for chain molecules consisting of 10 freely jointed segments, each of length 1 = 2.5 A. The end-to-end length r is in Angstrom units and W(x, y, z) is expressed in A". ... Fig. 77.—Gaussian density distribution of the chain displacement vec tors for chain molecules consisting of 10 freely jointed segments, each of length 1 = 2.5 A. The end-to-end length r is in Angstrom units and W(x, y, z) is expressed in A". ...
The introduction of vectors of constant displacement length to represent the individual elements, which actually vary in length, is rendered more plausible by inquiry into the effect of incorporating this artifice in the treatment of the freely jointed chain. In this case V = m H. Upon substitution of this expression together with n nlm in Eq. (17), the previous expression for / , Eq. (6), is recovered. Hence the calculated distribution is unaff ected by an arbitrary subdivision of the chain in this manner. We conclude that the value chosen for m in the reduction of the real chain to an equivalent freely jointed chain likewise is inconsequential (within the limits on m stated above). [Pg.412]

Parameter characterizing the Gaussian distribution of the end-to-end (displacement) length of a polymer chain (Chap. X et seq.). [Pg.649]

Fig. 8. Radial distribution function W(r) of the chain displacement vectors for chain molecules consisting of lO freely jointed segments, each of length I = 0.25 nm W(r) is expressed in nm and r in nm (3). Courtesy of Cornell University Press. Fig. 8. Radial distribution function W(r) of the chain displacement vectors for chain molecules consisting of lO freely jointed segments, each of length I = 0.25 nm W(r) is expressed in nm and r in nm (3). Courtesy of Cornell University Press.
In the ideal case, when one considers the network of chains of equal lengths, the stresses under the given deformation can be obtained in a very simple way. In virtue of the speculations of the previous section, free energy of the whole network can be represented as the sum of free energy of all the chains, while each of the equal chains of the network can be characterised by the same equilibrium distribution function W(s), where s is the separation between adjacent junctions. In the state without deformation, the function has the form (1.5), while in a deformed state, it depends on the displacement gradients (1.39). The free energy of the whole network can be written down simply as... [Pg.18]

A hydrocarbon chain is in a constant thermal motion, and without external force field, the chains fluctuate around the most stable position given by the distribution of possible conformations at the temperature. The action of external forces at the ends of a molecule causes displacements of chains from their equilibrium conformations and evokes retractive forces. For a hydrocarbon chain of M = 14,000, extended length 125.5 nm, and the end-to-end distance r = 1 mn, the maximum exerted force is 10 MPa. The level of forces exerted by the random coil macromolecules are much lower than the theoretical strength of the primary bonds. The presence of strong intermolecular interactions, such as hydrogen bonds in polyamides, affects the retractive force substantially, causing a restriction of the number of possible chain conformations. In addition, the transitions... [Pg.411]


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