Kuhn’s length

Suppose that the molecular geometry r, corresponds to a linear chain with n atoms and that we use the span R for characterization. Moreover, let / be the mean separation between bonded atoms, known as Kuhn s length The chain can be folded in many different ways, depending on its accessibility to low and high energy conformations ... 

If we retain only conformations in which the chain lies in almost perfect lines, it is evident that the average of R over those conformations, R), will scale linearly with R) nl. [The symbol is reserved here for a scaling relation. The relevant part of the scaling is the dependence with n the Kuhn s length provides only the units of length.]... 

Factors such as the valence angles or the hindrance to the free rotation contribute to the stretching of the chain which can thus be represented as nk units of length Lk linked together (Figure 5.8). Obviously, n and (or Kuhn s length) do not... 

Such representation gives an idea of the rigidity of the chain. The latter is more flexible if Ljt is small and close to L. Macromolecules whose Kuhn s length is large (and nk small compared to n) cannot be viewed as random coils but rather as worm-like chains. 

Here, b denotes the Kuhn s statistical segment length. The network is represented by a huge chain internally cross-linked at cross-linking points where it touches and at the surfaces of Mf filler particles. The point-like local cross-hnk constraints are easy to handle and can be represented by the term... 

The decrease in amount of bound iodine with increase in iodide-ion concentration is reflected in a decrease in the wave length of maximum absorption of the complex.166 On the basis of Kuhn s theory for polyenes,171 this has been related to the breaking of resonating chains of iodine molecules.172... 

Kirkwood and Riseman (1948) did not encounter this problem, because they used the bead-rod or, in other words, pearl-necklace model of macromolecule (Kramers 1946), in which A is a number of Kuhn s stiff segments, so that N present the length of the macromolecule. 

All carbon-carbon bonds in the skeleton have 50% double bond character. This fact was later confirmed by X-ray diffraction studies. A simple free-electron model calculation shows that there is no energy gap between the valence and conduction bands and that the limit of the first UV-visible transition for an infinite chain is zero. Thus a simple free-electron model correctly reproduces the first UV transition with a metallic extrapolation for the infinite system. Conversely, in the polyene series, CH2=CH-(CH=CH) -CH=CH2, he had to disturb the constant potential using a sinusoidal potential in order to cover the experimental trends. The role of the sinusoidal potential is to take into account the structural bond alternation between bond lengths of single- and double-bond character. When applied to the infinite system, in this type of disturbed free-electron model or Hiickel-type theory, a non-zero energy gap is obtained (about 1.90 eV in Kuhn s calculation), as illustrated in Fig. 36.9. 

It is striking to note that bond-length equalization occurs in the middle of a soliton. The soliton can, therefore, be viewed as a kind of metallic island on an otherwise semiconducting alternating chain in agreement with Kuhn s original ideas, presented above. 

Table 56 Throughputs (grams per 24 h) of Kuhn s multiple-tube column, having 100 tubes, diam 0.2 cm, length 2 m ...

Hence, the perfonned within the framewoik of fractal approach analysis of behavior of polystirene, modified by Dendron s, in diluted solutions gave the same conclusions, as the analysis within the fiamewoik of classical approaches. The main distinction of the indicated approaches is the fact, that the structural model, allowing to describe quantitatively macromolecules structural state and conformation, was placed in the fractal approach base. Other characteristics (gyration radius, Kuhn segment length and so on) are the function of the indicated structural state of a macromolecule. The fractal analysis methods, used for the description of linear flexible-chain polymers behavior, can be applied successfiilly also in case of polymers with more complex macromolecular architecture. 

A is the step length of Ising s lattice which contains macromolecules (see Figure 1.23d). The short-range interewition is included into A, therefore it is equivalent to Kuhn s segment. 

In the model of a continuous chain of length IV, the microscopic state of the chain is described by the conformation c(t) (0 < t < N), with c(t) denoting the spatial location of the point at a dimensionless distance t from the chain s end (Freed, 1972). Moreover, c(r) = (d//) f(T) is accepted, where I is an analogue of Kuhn s segment. This is introduced to simplify the Hamiltonian of the coil. 

Thus, the effective interaction has the form of dficay (serwning), and f is the screening length in the units of Kuhn s segment (cf. Equation 5.1 256)... 

Kawaguchi et al. 1998). Hence, one may conclude that the process of repetition at the PS-PS interfaces is controlled by an elementary act of a-relaxation (conformational transitiorrs in the chain backbone), and the persistence length of this kinetic rmit is eqtral to statistical Kuhn s segment. Second, both the values of E (D) at the interface and of E ( )"" are smaller than those of E (D) in the brrlk (Jou, 1986) and E (a) " (Bershtein et al., 1994) by a factor of 2. It indicates that the decrease in at flee polymer... 

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