Kuhn segment number

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]...

FIGURE 22 The relation between Kuhn segments number per macromolecule N on parameter corresponding to the Eq. (8), for PBA (1), PPTA (2) and PA (3). The corrected values were used. 

The Rouse model describes the dynamical properties of melts of macromolecules of a relatively small number of Kuhn segments, Ncritical number Nc is the number of Kuhn segments for the critical molecular mass Me- Flexible polymers have critical Kuhn segment numbers typically in the range Mc=40- 60 [1, 42-44, 52]. On the other hand, chain dynamics in concentrated systems of polymers with N Nc is much slower than expected on the basis of the Rouse model. Alluding to chain entanglements that are considered to become relevant in this case, one speaks of entangled dynamics. For example, experimental terminal relaxation times and center-of-mass self-diffusion coefficients scale as and... 

The parameter Cr expresses the effectiveness of the hindrance release by segment fluctuation and varies between 0 and 1. An empirical expression for it as a function of the Kuhn statistical segment number N is given in Sect. 8. Although Cr contributes only to the correction terms in de/Le, fr(de/Le) changes from 1 to 2.56 in the range of allowable values of de/Le. Thus, the factor fr (de/Le) is more important than the factor f (de/Le) in Dx. 

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. 

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 pressure p includes both the partial pressure of the gas of Brownian particles n(N +1 )T and the partial pressure of the carrier monomer liquid. We shall assume that the viscosity of the monomer liquid can be neglected. The variables xt k in equation (9.19) characterise the mean size and shape of the macromolecular coils in a deformed system. The other variables ut k are associated mainly with orientation of small rigid parts of macromolecules (Kuhn segments). As a consequence of the mesoscopic approach, the stress tensor (9.19) of a system is determined as a sum of the contributions of all the macromolecules, which in this case can be expressed by simple multiplication by the number of macromolecules n. The macroscopic internal variables x -k and u"k can be found as solutions of relaxation equations which have been established in Chapter 7. However, there are two distinctive cases, which have to be considered separately. 

The process was simulated for the copolymer of the structure I with n = 1, 5 and 10 in the structural unit of dimethylsiloxane chain, and the Kuhn segment A and /nM0 were calculated, where MO is the molecular mass of the structural unit, is the mean-square distance between chain ends at free rotation around virtual bonds 1, n is the number of structural units. 

It does not affect the exponents in the equation (i.e.. the dependence of on N), however, but simply introduces a prefactor. This suggests a different approach, where we consider the number of adjacent bonds whose combinations of allowed rotations essentially behave like a freely jointed unit when taken collectively. We would then have Nx effective segments each of length lp known as the Kuhn segment length (Figure 8-36), defined in Equation 8-12 ... 

Kuhn was the first to point out that the dimensions of a chain with given persistence p may always be described as if it were completely flexible (see (5.1.1)) by grouping a number of monomer units together into statistical chain elements (s.c.e.) or Kuhn segments. The number a of bonds in such an s.c.e. is the larger the stlffer the chain. The basic idea is that such s.c.e. s may be considered as orlentatlonally independent they are then independent subsystems as defined in sec. 1.3.6. The real chain of N bonds is now modelled as an equivalent ideal chain of = N/a s.c.e. s and the Kuhn length becomes bt (where a > 1, b > 1). Then (r ) = vdilch equals = 6pN(, provided that a... 

For EC, Bheda, Fellers and White find Vp/Vp = 1.43 in dichlordacetic acid and 1.18 in acetic acid. For HPC, they report v /vp = 1.29 to 1.37 in four different solvents. These results are compatible with the theory for monodisperse rods. It is to be noted that polydispersity of segment lengths should be unimportant if the chains are uniform in structure and composition or, more generally, if dissipation of directional persistence over the span of a Kuhn segment is the result of a number of restricted changes in chain direction. Hence, corrections for polydispersity should not be required for chains conforming to the representation here adopted. 

Hence, the theory permits the determination of XA = SX (where S = A/X is the number of monomer units in the Kuhn segment) by utilizing the experimental dependence of [nl on M, or of f (i. e. D or Is ) on M). Consequently, to find the parameters of equilitoium chain rigidity, A or S, from hydrodynamic data, it is also necessary to know the value of X, Wl n this condition is fulfilled, the chain diameter, d, can also be determined. 

The flexibility of the chain is caused by a rotation about the O-C and 0-C bonds between neighboring glucose rings. If a real chain is replaced by an equivalent chain each unit of which consists of two parallel A/2 bonds about which rotation is possible and one 6 bond (normal to the two first bonds) about which no rotation takes place, it can be shown that the number of monomer units Sf in the Kuhn segment of a cellulose chain with unhindered rotation is given by... 

Fig. 50. Number of monomer units Si 2 in a Kuhn segment of copolymer molecules of cyclohexanamide and caprolactam vs. relative content Z of cyclohexanamide units in the chain according to flow birefringence data Points and Curve 1 experimental data Curves 2 and 3 plotted according to Eqs. (74) and (75)...

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