Statistical chain segments

The stress-optical coefficient, Ka, of an elastomer network is a constant, independent of extension ratio and crosslink density. It is directly proportional to the difference between the longitudinal and transverse polarizabilities of the statistical chain segment (fei — 2) ... 

Substitution of the experimental values for as and 0 which is estimated by the Shultz-Flory method as listed in Table 8, into Eq. (31) permits to determine d. The d values determined on the basis of the statistical chain segment model by these two methods (Eq. (30) and Eq. (31)) are quite consistent (Table 12). The use of X = 20 (very weak draining effect) in Eq. (30) yields an unreasonably large d value, nearly 30 times larger than the d value (24.6 A) obtained by the thermodynamic method. This indicates that the draining effect cannot be neglected. 

Number of statistical chain segments between crosslinks Number of statistical drain segments between entanglements Probability of crystalline sequence linking two adjacent lamellae Hydrostatic pressure Amorphous phase solubility... 

For a simple rubber-like network, the maximum draw ratio varies with the number of statistical chain segments between crosslinks as... 

Smith, Lemstra and Booij developed a similar argument for a gel-spun fibre. The molecular weight between entanglements in the undiluted polymer melt is denoted by M,. The number of statistical chain segments between entanglements is given by... 

Length of the i th type of bond along the shortest path across the chain backbone. Length of a statistical chain segment (Kuhn segment). 

Some important properties of polymer chains in dilute solutions [steric hindrance parameter, characteristic ratio, persistence length, radius of gyration, statistical chain segment length (introduced earlier, in Chapter 11), intrinsic viscosity, and viscosity at small but finite concentrations] will be discussed, and new correlations will be presented for the steric hindrance parameter and the molar stiffness function, in Chapter 12. 

Figure 11.15. Effects of input material parameters on stress-strain curves of elastomers under uniaxial tension, as calculated by the theory of rubber elasticity with finite chain extensibility. G denotes the shear modulus, while n denotes the average number of statistical chain segments (Kuhn segments) between elastically active network junctions, (a) Engineering stress a as a function of draw ratio X, as calculated by using Equation 11.41. (b) True stress (simply equal to aX for an elastomer) as a function of true strain [In (A,)].

Several parameters, most of which are interrelated and can be estimated in terms of each other, are utilized to describe the conformational properties of polymer chains [1,2]. These quantities include the steric hindrance parameter a, the characteristic ratio Cx, the persistence length Ip, the statistical chain segment (or Kuhn segment) length lk, the root mean square radius of gyration Rg (often briefly referred to as simply the "radius of gyration"), and the molar... 

Table 12.1. Literature data for the statistical chain segment length lk, in units of A, tabulated by Bershtein and Yegorov [18], The average value is listed below for any polymer for which more than one lk value or a range of lk values was indicated.

Nc Number of statistical chain segments between crosslinks... 

A somewhat more sophisticated approach to domain formation and fine structure was taken by Meier (1969,1970). As with Inoue et al. (1970a,f ) and Krause (1969,1970,1971), the A-B junction was restricted to a location somewhere in the interfacial region. Meier s model (Figure 4.11) assumes that random flight statistics and regular solution theory hold, that statistical chain segments (not block lengths) are of equal size, and that chain perturbation is characterized by the usual parameter a ... 

In the systems of flexible chain polymers, asymmetric single elements are identified with the uniaxial statistical chain segments. Considering a cylindrical cluster of uniaxial elements oriented at 0 = (0, ), the condition of compliant orientation reduces fraction of the chain segments involved in the association process to w(6))A where... 

In Eq. 3, n denotes the number of statistical chain segments of length / in the strand. Also, is the inverse Langevin function and cr is a proportionality constant. 

It must be noted that the orientation given by our fluorescence polarization measurements refers to the anthracene group, the transition moment of which lies along the chain axis and is located in the middle of the chain. This particular position implies that the measured orientation reflects the orientational behaviour of statistical chain segments of the central part of the chain. It seems difficult to state the size of this central part precisely. Owing to the molecular weight of... 

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