Extensibility finite chain

Finite chain extensibility is the major reason for strain hardening at high elongations (Fig. 7.8). Another source of hardening in some networks is stress-induced crystallization. For example, vulcanized natural rubber (cw-polyisoprene) does not crystallize in the unstretched state at room temperature, but crystallizes rapidly when stretched by a factor of 3 or more. The extent of crystallization increases as the network is stretched more. The amorphous state is fully recovered when the stress is removed. Since the crystals invariably have larger modulus than the surrounding... 

It will be shown in Chapter 11 that the correlations developed in this monograph can be combined with other correlations that are found in the literature (preferably with the equations developed by Seitz in the case of thermoplastics, and with the equations of rubber elasticity theory with finite chain extensibility for elastomers), to predict many of the key mechanical properties of polymers. These properties include the elastic (bulk, shear and tensile) moduli as well as the shear yield stress and the brittle fracture stress. In addition, new correlations in terms of connectivity indices will be developed for the molar Rao function and the molar Hartmann function whose importance in our opinion is more of a historical nature. A large amount of the most reliable literature data on the mechanical properties of polymers will also be listed. The observed trends for the mechanical properties of thermosets will also be discussed. Finally, the important and challenging topic of the durability of polymers under mechanical deformation will be addressed, to review the state-of-the-art in this area where the existing modeling tools are of a correlative (rather than truly predictive) nature at this time. 

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,)].

At very high deformations, the main contribution to reinforcement comes from the change of network chain statistics due to finite chain extensibility, but supported here with filler particle as... 

For fast flow deformations of polymer fluids, a non-linear theory of chain deformation and orientation is considered. To account for non-hnear effects and finite chain extensibility, inverse Langevin chain statistics is assumed. Time evolution of chain distribution function in the systems with inverse Langevin chain statistics has been discussed in earlier papers [12,13] providing physically sensible stress-orientation behaviour in the entire range of the deformation rates and chain deformations. 

Horgan CO, Ogden RW, Saccomandi G (2004) A theory of stress softening of elastomers based on finite chain extensibility. Proc R Soc A 460 1737-1754... 

Comparison of Figures 1 and 2 shows that for swelling degrees greater than about ten, finite chain extensibility must be accounted for through non-Gaussian chain statistics. While such theoretical calculations have been proven to be qualitatively... 

An additional slow process of chain extension underneath the obstructing overgrowth has also been considered in [49]. The second half of the chain was allowed to attach to the half-chain nearest to the extended-chain substrate at rate D. This process, mimicking the sliding diffusion , preserved a finite growth rate even at the rate minimum, i.e. it made the minimum shallower. 

Although equation (6-12) adequately describes the behavior of an elastomer at high extensions (> 10% for natural rubber), it is nevertheless true that the coefficient dH/dL)r,p has a finite value and cannot be neglected in a complete treatment of rubber elasticity. [It should be stated that equation (6-12) is inadequate to describe the behavior of most elastomers at very high extensions because of increasing limitations on chain extension and motion, as well as the onset of crystallization in many elastomers. When this occurs, the term (dH/dL)r,p once again becomes important and may actually outweigh the term T(dS/dL)r,p. ... 

It is presently a known fact that, while the Raman frequencies of each individual short finite-chain oligoene do not show "dispersion" (i.e. change of Raman frequencies with change of the wavelength of the exciting radiation), any sample of trans-PA shows, remarkably, die phenomenon of "Raman dispersion." This fact has been the subject of extensive and highly controversial theoretical and experimental studies by many authors, but we shall limit ourselves to the basic facts, and to the consequences in stractmal and dynamical studies of trans-PA [25]. 

The structure of the simulated block copolymer systems has been characterized in detail [38-40]. Temperature dependencies of various structural parameters have shown that all of them change in a characteristic way in correspondence to Tqdt- The microphase separation in the diblock copolymer system is accompanied by chain extension. The chains of the diblock copolymer start to extend at a temperature well above that of the transition to the strongly segregated regime. This extension of chains is related also to an increase of local orientation correlations, which appear well above the transition temperature. On the other hand, the global orientation correlation factor remains zero at temperature above the microphase separation transition and jumps to a finite value at the transition. 

S cm which is well below the Mott minimum value, the conductivity at 1 K can be around 4 S cm . Probably, the structural disorder in PEDOT samples is considerably less than that in other systems. Since the /3-positions in the thiophene rings are blocked by the ethylenedioxy group, the chain extension is possible only through the a-positions. A large finite conductivity ( 150 S cm ) has been observed in metallic PEDOT and PMeT systems yet, these systems are just on the metallic side of the M-I transition. 

We may conclude from this result that in treatments of polymer chains in solution or the melt, use of the Gaussian distribution function is always permitted. However, situations exist which require the application of the exact equation. Rubber elasticity, or treatments of yielding properties of polymers are affected by the limits in extensibility given for finite chains. Here, the Gaussian approximation can be used for small deformations only, and dealing with large deformations necessitates an introduction of the exact expression. 

Expanded chains are found in dilute solution in good solvents. The effective interaction energy between two monomers is always repulsive here, and, as a consequence, chains become expanded. Expansion will come to an end at some finite value since it is associated with a decreasing conformational entropy. The reason for this decrease is easily seen by noting that the number of accessable rotational isomeric states decreases with increasing chain extension. The decrease produces a retracting force which balances, at equilibrium, the repulsive excluded volume forces. 

In the case of longitudinal gradient, the viscosity increases very rapidly and goes to infinity at 3 = -5 for a coil with unlimited extensibility. Finite chain length limits the increase up to a limiting value L/a M. The rapid decrease... 

Using both condensation-cured and addition-cured model systems, it has been shown that the modulus depends on the molecular weight of the polymer and that the modulus at mpture increases with increased junction functionahty (259). However, if a bimodal distribution of chain lengths is employed, an anomalously high modulus at high extensions is observed. Finite extensibihty of the short chains has been proposed as the origin of this upturn in the stress—strain curve. 

Complicated theories of ionic gel swelling [99, 113, 114] must inevitably take into account the real electrostatic interactions, the finite extensibility of chains, as well as the electrostatic persistence length effect. Their application is most advisable in the case of strongly charged hydrogels [114]. 

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